Book EVT · English

Index of statements

The 644 statements of Topological Vector Spaces, by chapter and §. Pick a kind, filter by a word, or put the most cited first.

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I. Topological vector spaces over a valued division ring

§ 1. Topological vector spaces

Definition 1 — Given a topological division ring K (GT, III, § 6.7) and a set E such that E has 1^∘ the structure of a left vector space on K; 2^∘ a topology compatible with the structure of th… 077V
Example — 1) If E is a left vector space over a discrete topological division ring K, the discrete topology on E is compatible with the vector space structure of E (this is not so if K is… 077W
Proposition 1 — For all α ∈ K and every point b ∈ E, the mapping x ↦ α x + b of E in itself is continuous. Further, if α ≠ 0, this mapping is a homeomorphism of E on itself. 077X
Corollary — If A is an open (resp. closed) set in E, then α A is open (resp. closed) in E for every α ≠ 0 in K. 077Y
Example Let I be an arbitrary set of indices; it is known (GT, X, § 3.2) that a norm |x| can be defined, on the set of bounded mappings x = (ξᵢ) of I in K, B(I; K) (also written B_K(I) or… 077Z
Remark Given a closed vector subspace M of a Hausdorff topological vector space E, it is not necessarily the case that there exists an (algebraic) complementary vector subspace to M that… 0780
Remark 1 If B is a precompact set of K, then for every neighbourhood V of 0 in E, there is a neighbourhood U of 0 in E such that BU ⊂ V. For, if W is a neighbourhood of 0 in E such that W +… 0781
Remark 2 From 1) it follows that, if B is a precompact set of K and H is a precompact set of E, then the mapping (λ, x) ↦ λ x restricted to B × H is uniformly continuous. For, if V is a nei… 0782
Definition 2 A complete normed space on a non-discrete valued division ring is called a Banach space. 0783
Example If K is a non-discrete valued division ring then the space B(I ; K) (I, p. 4, Example) is complete (GT, X, § 3.1, cor. 1). This is also true for the space ℓ_K¹(I) (I, p. 4, Example… 0784
Definition 3 Let K be a valued division ring and E a left vector space over K; we say that a subset M of E is balanced if, for all x ∈ M and all λ ∈ K such that |λ| ≤ 1, it is true that λ x ∈ M… 0785
Proposition 2 In a topological vector space E over a valued division ring K, the closure of a balanced set M, is a balanced set. 0786
Proposition 3 Let K be a valued locally compact and non-discrete division ring and E be a Hausdorff topological vector space (resp. a topological vector space) over K. For every compact (resp. p… 0787
Definition 4 Let K be a non-discrete valued division ring and E be a left vector space on K with two subsets A and B. We say that A absorbs B if there exists α 0 such that λA ⊃ B for every λ ∈… 0788
Proposition 4 In a topological vector space E on a non-discrete valued division ring K there exists a fundamental system 𝔅 of closed neighbourhoods of 0 such that : (EV₁) Every set V ∈ 𝔅 is bala… 0789
Remark 1 In a normed space on a non-discrete valued division ring the set of open balls (resp. closed balls) with centre 0 is a fundamental system of neighbourhoods of 0 which satisfy the c… 078A
Remark 2 When the division ring of scalars K is the field \mathbf{R} or the field \mathbf{C}, every filter base \mathfrak{B} on E which satisfies just the two axioms (EVI) and (EVIII) is a… 078B
Remark 3 If K is a discrete division ring, conditions (EVT'I) and (EVT'III) are true for any topology on E. Arguing as in prop. 4, one easily sees that if E is a topological vector space on… 078C
Proposition 5 — Let Eᵢ (1 ≤ i ≤ n) and F be topological vector spaces on a non-discrete valued field K. In order that a multilinear mapping f of ∏ᵢ₌₁ⁿ Eᵢ in F should be continuous in the product… 078D
Proposition 6 With the same hypotheses on Eᵢ (1 ≤ i ≤ n) and on F as in prop. 5, in order that a set E of multilinear maps of ∏ᵢ₌₁ⁿ Eᵢ in F be equicontinuous it is sufficient that the set be equ… 078E
Proposition 7 Let (Eᵥ)_v ∈ I be a family of topological vector spaces on a topological division ring K. Let E be a vector space on K and for each v ∈ I, let fᵥ be a linear mapping of E in Eᵥ. Th… 078F
Lemma — Let M and N be two vector spaces, and g a linear mapping of M in N. If T₀ is a topology compatible with the vector space structure of N, then the inverse image of T₀ by g is comp… 078G
Corollary 1 Let G be a topological vector space on K. In order that a set H of mappings of G in E be equicontinuous, it is necessary and sufficient that, for all i ∈ I, the set fᵢ ∘ u where u… 078H
Corollary 2 If the spaces Eᵢ are Hausdorff, then in order that T be Hausdorff, it is necessary and sufficient that, for every x ≠ 0 in E, there should exist an index i ∈ I, such that fᵢ(x) ≠ 0… 078I
Corollary 3 Suppose the Eᵢ are complete and φ(E) is closed in F = ∏_i ∈ I Eᵢ. Then E is complete in the topology T. 078J
Example — Let D'(R) be the space of distributions on R; for p a number such that 1 ≤ p ≤ +∞, let j : Lᵖ(R) → D'(R) be the canonical injection, which is continuous (when Lᵖ(R) carries its n… 078K
Corollary 4 Let E be a vector space over a topological division ring K, and let (Tᵢ)_i ∈ I be a family of topologies compatible with the vector space structure of E; then the upper bound T of… 078L

§ 2. Linear varieties in a topological vector space

Proposition 1 — In a topological vector space E, the closure of a linear variety is a linear variety. 0790
Corollary — In a topological vector space E, every hyperplane is either closed or everywhere dense. 0791
Definition 1 — A set A, in a topological vector space E, is total if, and only if, the closed vector subspace generated by A coincides with E (i.e. the set of linear combinations of elements of… 0792
Example — 1) In the normed space C(I; C) (on the field C) of functions, continuous on I = 0, 1, with values in C, the restrictions to I of the functions xⁿ \ (n ∈ N) form a total set, by t… 0793
Definition 2 — A family (aᵢ)_i ∈ I of points of a topological vector space E is called topologically independent if for any κ ∈ I, the closed vector subspace generated by the aᵢ, with i ≠ κ, do… 0794
Example 3 In the normed space C(I; C) of continuous functions defined over I = 0, 1, the restrictions to I of the functions e^2nπ i x (n ∈ Z) form a topologically independent family. If f(x)… 0795
Example 4 In the normed space C(I; C) of functions that are continuous over I = 0, 1, the restriction to I of the functions xⁿ (n ∈ N) form an algebraically independent family. But there exi… 0796
Remark 1 The family of topologically independent sets of a topological vector space is not necessarily inductive for the relation of inclusion (I, p. 25, exerc. 2); this situation is thus d… 0797
Remark 2 Let M be a closed vector subspace of E and (dotaᵢ)_i ∈ I a topologically independent family in the quotient space E/M. If aᵢ is any element of the class dotaᵢ, then from def. 2, an… 0798
Proposition 2 Every Hausdorff topological vector space E of dimension 1 over a non-discrete valued division ring K is isomorphic to Kₛ; in fact, for every a ≠ 0 in E, the mapping ξ ↦ ξ a of Kₛ o… ↩ 1 0799
Corollary 1 In a Hausdorff topological vector space E, over a non-discrete valued division ring K, every vector subspace D of dimension 1 is isomorphic to Kₛ. 079A
Corollary 2 Let E be a topological vector space over a non-discrete valued division ring. Every vector subspace D (of dimension 1) which is the algebraic complement of a closed homogeneous hyp… 079B
Theorem 1 Let E be a topological vector space over a non-discrete valued division ring. Let H be a hyperplane in E defined by the equation f(x) = α where f is a linear form not identically z… 079C
Corollary Every continuous linear form on E that is not identically zero is a strict morphism of E on Kₛ. 079D
Remark — There are examples of normed topological vector spaces over a complete non-discrete valued division ring, in which every continuous linear form is identically zero (I, p. 25, exe… 079E
Theorem 2 Every Hausdorff topological vector space E, of finite dimension n, over a complete non-discrete valued division ring K, is isomorphic to Kₛⁿ; in fact, for every basis (eᵢ)_1 ≤ i ≤… 079F
Corollary 1 In a Hausdorff topological vector space E over a complete non-discrete valued division ring K, every vector subspace F of finite dimension is closed in E. For, if F is of dimension… 079G
Corollary 2 Let K be a complete non-discrete valued division ring, and E be a Hausdorff topological vector space of finite dimension over K. If F is any topological vector space over K, then e… 079H
Corollary 3 In a Hausdorff topological vector space E, over a complete non-discrete valued division ring, every finite independent set is topologically independent. 079I
Corollary 4 Let E be a topological vector space over a complete non-discrete valued division ring. If M is a closed vector subspace of E and F is a vector subspace of E of finite dimension, th… 079J
Proposition 3 Let E be a topological vector space over a complete non-discrete valued division ring K. Let M be a closed vector subspace of finite codimension n in E. Then every subspace N that… 079K
Corollary Let E and F be two topological vector spaces over a complete non-discrete valued division ring. If F is Hausdorff and of finite dimension, then every continuous linear mapping of E… 079L
Remark — The results of Nos 2, 3 are no longer valid when K is discrete. For example, let K₁ be a non-discrete valued division ring and K be the discrete division ring obtained by endowin… 079M
Theorem 3 Let K be a complete non-discrete valued division ring. If E is a Hausdorff topological vector space over K, which is such that some neighbourhood V of 0 in E is precompact, then E… 079N
Lemma 1 — Any compact topological vector space E over a non-discrete valued division ring, is just the set 0. 079O
Remark The result of th. 3 is no longer true if K is a discrete division ring as is shown by the example of R (with the usual topology) considered as a topological vector space over the d… 079P

§ 3. Metrisable topological vector spaces

Remark One of the most important classes of metrisable vector spaces are the normed spaces (I, p. 3). But it must be noted that there exist metrisable vector spaces whose topology cannot… 07A5
Theorem 1 (Banach) Let E and F be two metrisable vector spaces over a non-discrete valued division ring K, and let u be a continuous linear mapping of E in F. Suppose that E is complete. Then the fol… 07A6
Lemma 1 Let E and F be two topological vector spaces over a non-discrete valued division ring K, and let u be a continuous linear mapping of E in F such that the image of E is not meagre… 07A7
Lemma 2 Let E and F be two metric spaces, and suppose that E is also complete. Let u be a linear mapping of E in F having the following property: whatever the number r > 0, there exists a… 07A8
Corollary 1 If E and F are two complete metrisable vector spaces over a non-discrete valued division ring, then every bijective continuous linear mapping of E on F is an isomorphism. 07A9
Corollary 2 Let E be a vector space over a non-discrete valued division ring, let T₁ and T₂ be two topologies on E compatible with its vector space structure and for each of which E is metrisa… 07AA
Corollary 3 Let E and F be two complete metrisable vector spaces over a non-discrete valued division ring. In order that a continuous linear mapping u of E in F should be a strict morphism, it… 07AB
Corollary 4 Let E be a complete metrisable vector space over a non-discrete valued division ring. If M and N are two closed vector subspaces, that are (algebraic) complements in E, then E is t… 07AC
Corollary 5 (The closed graph theorem) — Let E and F be two complete metrisable vector spaces over a non-discrete valued division ring. In order that a linear mapping u of E in F be continuous, it is necessary and suffi… 07AD
Example — Let E be a vector subspace of the space of real-valued functions defined on I = [0, 1]; let | f | be a norm on E, under which E is complete, and such that its topology is finer t… 07AE
Proposition 1 Let E, F be two topological vector spaces over a non-discrete valued division ring K. We suppose that : 07AF

II. Convex sets and locally convex spaces

§ 1. Semi-norms

Definition 1 Let E be a left vector space over K. A mapping p of E in R₊ = 0, + ∞ , is called a semi-norm on E if it satisfies the following axioms : (SNI) If x ∈ E and λ ∈ K then p(λ x) = |λ|… 07AP
Example 1 A norm on E is a semi-norm p such that the relation p(x) = 0 implies that x = 0 (I, p. 3). 07AQ
Example 2 For every linear form f on E, the function x ↦ |f(x)| is a semi-norm on E. 07AR
Example 3 If pᵢ (1 ≤ i ≤ n) is a finite set of semi-norms on E, then clearly p'(x) = sup_1 ≤ i ≤ n pᵢ(x) and p''(x) = ∑ᵢ₌₁ⁿ αᵢ pᵢ(x) (where the αᵢ are ≥ 0) are both semi-norms on E. 07AS
Proposition 1 Let E be a left topological vector space over K and let p be a semi-norm on E. The following conditions are equivalent : a) p is continuous in E. b) p is continuous at the point 0… 07AT
Corollary If p is a continuous semi-norm on E and q is a semi-norm such that q ≤ p, then q is continuous in E. 07AU
Remark — 1) The topology defined by a finite set of semi-norms pᵢ (1 ≤ i ≤ n) on E, can be defined by the single semi-norm p = sup_1 ≤ i ≤ n pᵢ. But a topology defined by an infinite set… 07AV
Example — Let C^∞(R) be the vector space over R of real valued functions that are infinitely differentiable in R. For every function and every pair of integers n ≥ 0, m ≥ 1, put 07AW
Proposition 2 On a vector space E, let T be the topology defined by a set of semi-norms Γ. 07AX
Proposition 3 Every Hausdorff topological vector space E over K, whose topology is defined by a set of semi-norms, is isomorphic to a sub-space of a product of Banach spaces. 07AY
Proposition 4 Let Eᵢ (1 ≤ i ≤ n) and F be topological vector spaces over K; we suppose that, for every i, the topology of Eᵢ is defined by a directed set of semi-norms Γᵢ, and that the topology… 07AZ
Corollary — The set H is equicontinuous if, and only if, for every semi-norm q ∈ Γ, there exists a neighbourhood of 0 in ∏ᵢ₌₁ⁿ Eᵢ, in which the functions q ∘ u, for u ∈ H, are uniformly boun… 07B0
Proposition 5 — Let E, F be two topological vector spaces over a non-discrete valued division ring K; suppose that the topology of E (resp. F) is defined by a set of semi-norms Γ (resp. Γ'). Let… 07B1
Corollary 1 — Suppose that T, T' are two topologies on a vector space E over K defined, respectively, by two sets of semi-norms Γ and Γ'. T is finer than T' if, and only if, for every semi-nor… 07B2
Corollary 2 — Suppose that the topology T of a topological vector space E over K is defined by a directed set of semi-norms Γ; for each semi-norm p ∈ Γ, let Eₚ be the space obtained from E usi… 07B3

§ 2. Convex sets

Definition 1 — A subset A of an affine space E is convex if, for every two points x, y of A, the closed segment with end points x, y is contained in A. 07B4
Example 1 Every linear affine variety of E (and in particular the empty set) is convex. 07B5
Example 2 The only non-empty convex sets in R are the intervals (GT, IV, § 2.4, prop. 1). 07B6
Example 3 Let E be a vector space and |x| a norm on E; the unit ball B, formed by the points x 07B7
Remark — Let A be a convex subset of a vector space E; for any scalars α > 0 and β > 0 we have α A + β A = (α + β) A. In other words, for any x ∈ A, y ∈ A, there exists z ∈ A such that (α… 07B8
Proposition 1 Let (xᵢ) be a family of points of a convex subset A; every barycentre ∑ᵢ λᵢ xᵢ of the xᵢ formed using positive masses λᵢ (such that ∑ᵢ λᵢ = 1 and λᵢ = 0 except for finitely many of… 07B9
Proposition 2 Let E and F be two affine spaces and f be an affine linear mapping of E in F; then the image of a convex subset of E under f, and the inverse image of a convex subset of F under f… 07BA
Proposition 3 In the affine space E, let H be a hyperplane defined by the relation g(x) = 0, where g is a non-constant affine function on E. Then the half-spaces defined by the relations g(x) ≥… 07BB
Proposition 4 The points of A, a convex subset of an affine space E are strictly on the same side of a hyperplane H if, and only if, A does not meet H. 07BC
Proposition 5 — The intersection of any family of convex subsets of an affine space E is convex. 07BD
Proposition 6 — Let (Eᵢ)_i ∈ I be a family of vector spaces, and for each i ∈ I, let Aᵢ be a non-empty subset of Eᵢ. Then the set A = ∏_i ∈ I Aᵢ is convex in E = ∏_i ∈ I Eᵢ, if, and only if, for… 07BE
Corollary — In the space Rⁿ every parallelootope (GT, VI, § 1.3) is a convex subset. 07BF
Proposition 7 — Let A and B be two convex subsets of the vector space E. For any real numbers α, β the set α A + β B (set of points of the form α x + β y, where x varies in A, and y in B) is con… 07BG
Definition 2 — Given a subset A of an affine space E, we call the intersection of all convex sets containing A, the convex envelope of A, that is to say (II, p. 9, prop. 5) it is the smallest c… 07BH
Proposition 8 — For any family (Aᵢ)_i ∈ I of convex subsets of an affine space E, the convex envelope of ⋃_i ∈ I Aᵢ is precisely the set of linear combinations ∑_i ∈ I λᵢ xᵢ, where xᵢ ∈ Aᵢ, λᵢ ≥… 07BI
Corollary 1 The convex envelope of a subset A of E is identical with the set of linear combinations ∑ᵢ λᵢ xᵢ, where (xᵢ) is any finite family of points of A, the numbers λᵢ > 0, for all i and… 07BJ
Corollary 2 Let f be an affine linear mapping of the affine space E in the affine space F; for each subset A of E, the convex envelope of f(A) is the image under f of the convex envelope of A. 07BK
Definition 3 A subset C of an affine space E is a cone with vertex x0 if C is invariant for all homotheties of centre x0 and ratio 0. 07BL
Proposition 9 A pointed convex cone C is proper if and only if the non-pointed cone C', which is the complement of 0 in C, is convex. 07BM
Proposition 10 A subset C of E is a convex cone if and only if C + C ⊂ C and λ C ⊂ C for all λ > 0. 07BN
Corollary 1 If C is a non-empty convex cone, the vector space generated by C is the set C - C (the set of points x - y where x, y vary in C). 07BO
Corollary 2 If C is a pointed convex cone, the largest vector subspace contained in C is the set C ∩ (-C). 07BP
Proposition 11 Let (Cᵢ)_i ∈ I be a family of convex cones in E; the convex cone generated by the union of the Cᵢ is identical with the set of points ∑_i ∈ J xᵢ, where J is any finite subset of I… 07BQ
Corollary — For any subset A of E, the convex cone generated by A, is identical with the set of linear combinations ∑_i ∈ J λᵢ xᵢ, where (xᵢ)_i ∈ J is any finite non-empty family of points o… 07BR
Proposition 12 — If A is a convex set in E, then the convex cone generated by A is identical with C = ⋃_λ > 0 λ A. ↩ 1 07BS
Remark — 1) With the hypotheses of prop. 12, if 0 ∉ A, then the cone C is non-pointed, thus C ∪ 0 is proper. 07BT
Example — On the space E = Rᴬ of all finite real-valued functions defined over A, the relation of order given by « for all t ∈ A, x(t) leqslant y(t) » is compatible with the vector space s… 07BU
Proposition 13 — (i) The set P, of elements geqslant 0, of a preordered vector space E, is a pointed convex cone. 07BV
Example \ Let H be a real Hilbert space; in the vector space L(H) of continuous endomorphisms of H, the positive hermitian endomorphisms form a proper pointed convex cone; this cone, there… 07BW
Proposition 14 In a topological vector space E, the closure of a convex set (resp. of a convex cone) is a convex set (resp. a convex cone with the same vertex). 07BX
Definition 4 For any set A of a topological vector space E, the intersection of all the closed convex sets containing A is called the convex closed envelope of A; it is the smallest convex clos… 07BY
Proposition 15 — Let Aᵢ (1 ≤ i ≤ n) be a finite number of compact convex sets in a Hausdorff topological vector space E. Then the convex envelope of the union of the Aᵢ is compact (and is, theref… 07BZ
Corollary 1 — In a Hausdorff topological vector space the convex envelope of a finite set is compact. 07C0
Corollary 2 — In a topological vector space E, the convex envelope of a finite set is precompact. 07C1
Proposition 16 — Let A be a convex subset, with at least one interior point x₀, of a topological vector space E. For any point x ∈ overlineA, every point of the open segment with end points x₀, x… 07C2
Corollary 1 — The interior overset∘A of a convex set overlineA, is itself a convex set; if overset∘A is not empty, then it coincides with the interior of overlineA, and overlineA is a convex s… 07C3
Corollary 2 — The interior dotC of a convex cone C, is itself a convex cone; if dotC is not empty then it coincides with the interior of overlineC, and overlineC is a pointed convex cone that… 07C4
Proposition 17 — In a topological vector space E, let A be a set with at least one interior point, and such that all its points lie on the same side of an hyperplane H. Then H is closed, the inte… 07C5
Corollary — Let P be a pointed convex cone, with at least one interior point, of the topological vector space E. Then each linear form f that is not identically zero on E, and is positive fo… 07C6
Remark In a topological vector space E, every convex set C is connected. In fact, if a ∈ C, then C is a union of segments with end point a and closed at a; these are connected and the res… 07C7
Example The space Rⁿ with its usual topology and the order structure that is the product of the order structure of its factors is an ordered topological vector space. On the other hand, fo… 07C8
Proposition 18 In an ordered topological vector space E, let H be a set directed by the relation ≤. If the section filter of H has a limit in E, then this limit is the upper bound of H. 07C9
Definition 5 Let X be a convex subset of the affine space E. A real-valued finite function, defined over X is convex (resp. strictly convex) if for any two distinct points x, y of X and any rea… 07CA
Example If f is an affine linear function on E, then f and f² are convex functions on E; this is obvious for f since f(λ x + (1 - λ) y) = λ f(x) + (1 - λ) f(y); on the other hand, if α = f… 07CB
Proposition 19 Let X be a convex set of the affine space E; and let f be a real-valued function defined over X. Denote the set of points (x, a) ∈ E × R for which x ∈ X and f(x) leqslant a (resp… 07CC
Corollary If f is convex in X, then for all α ∈ R, the set of x ∈ X such that f(x) leqslant α (resp. f(x) < α) is convex. 07CD
Proposition 20 Let f be a convex function, defined over a convex set X of the affine space E. Then for every family (xᵢ)_1 ≤ i ≤ p of p points of X and every family (λᵢ)_1 ≤ i ≤ p of p real numbe… 07CE
Proposition 21 Let f be a convex function, defined over the non-empty open convex set X in the topological vector space E. Then f is continuous if, and only if, it is bounded above when restricte… 07CF
Corollary Every convex function f defined over an open convex set X in Rⁿ is continuous in X. 07CG
Proposition 22 Let A be a convex set, containing 0, in the vector space E. For all x ∈ E, put 07CH
Proposition 23 Let E be a topological vector space. If A is an open convex set which contains 0, then p_A is finite and continuous, and A = W(p_A, 1). If A is a closed convex set containing 0, th… 07CI

§ 3. The Hahn-Banach Theorem (analytic form)

Proposition 1 — Let E be a preordered vector space and V be a vector subspace of E such that every element of E is bounded above by an element of V. Given a linear form f on V that is positive f… 07DQ
Corollary In a topological vector space E with a compatible preorder structure, let P be the set of elements ≥ 0 in E. Let V be a vector subspace of E containing at least one interior point… 07DR
Theorem 1 (Hahn-Banach) Let p be a sub-linear function on a vector space E. Let V be a vector subspace of E and f a linear form on V such that, for all y ∈ V, we have f(y) ≤ p(y). Then there exists a line… 07DS
Corollary 1 — Let p be a semi-norm on the vector space E. Let V be a vector subspace of E and f a linear form on V such that |f(y)| ≤ p(y) for all y ∈ V. Then there exists a linear form h defi… 07DT
Corollary 2 — Let p be a semi-norm on the vector space E. Given a point x₀ ∈ E, there exists a linear form f defined over E, such that f(x₀) = p(x₀) and that |f(x)| ≤ p(x) for all x ∈ E. 07DU
Corollary 3 — Let V be a vector subspace of the normed space E and let f be a continuous linear form over V; then there exists a continuous linear form h defined over E which extends f and is… 07DV

§ 4. Locally convex spaces

Definition 1 — A topological vector space is locally convex (real) if there exists a fundamental system of neighbourhoods of 0 that are convex sets. 07E4
Proposition 1 — Let 𝔖 be a filter base on a vector space E formed from sets that are absorbent, symmetric and convex. Then the set 𝔅 of transforms of the sets of 𝔖 by homotheties of ratio > 0 is… 07E5
Corollary — The topology T of a topological vector space E, is defined by a set of semi-norms (II, p. 3) if, and only if, T is locally convex. 07E6
Proposition 2 — Let f be a continuous linear form defined over a vector subspace M, of a locally convex space E; then there exists a continuous linear form h that is defined over E and is an ext… 07E7
Remark — If g is a continuous linear mapping of M in the product space Rᴵ, then there exists a continuous linear mapping h of E in Rᴵ that is an extension of g; for writing g = (gᵢ), wher… 07E8
Corollary 1 — Let E be a locally convex space. If x₀ ∈ E is not in the closure of 0, then there exists a continuous linear form f defined over E with f(x₀) ≠ 0. 07E9
Corollary 2 — Let M be a finite dimensional vector subspace of E, a locally convex Hausdorff space. Then there exists a closed vector subspace N of E, which is the topological complement of M… 07EA
Proposition 3 — In a locally convex space E, the balanced convex envelope of a precompact set is itself a precompact set. 07EB
Corollary — If, in a locally convex Hausdorff space E, a compact set X is contained in a complete convex set (complete in the uniform structure induced by that of E) then the convex closed e… 07EC
Proposition 4 Let E be a vector space and let (Eᵢ)_i ∈ I be a family of locally convex spaces. For each i ∈ I, let fᵢ be a linear mapping of E in Eᵢ; then the topology T on E, which is the coars… 07ED
Proposition 5 Let E be a vector space, and (F_α)_α ∈ A be a family of topological vector spaces and for each α ∈ A, let g_α be a linear mapping of F_α in E. 07EE
Corollary 1 With the notations of prop. 5, let H be a set of linear mappings of E in the locally convex space G. Suppose that E is the sum of its subspaces g_α(F_α); then H is equicontinuous f… 07EF
Corollary 2 Let (J_λ)_λ ∈ L be a partition of the index set A. Let (G_α)_α ∈ A be a family of locally convex spaces and (F_λ)_λ ∈ L be a family of vector spaces. For each λ ∈ L, let h_λ be a l… 07EG
Example — Let X be a locally compact space and E = K(X; R) the vector space of finite continuous real valued functions defined over X with compact support. For every compact subset K of X… 07EH
Remark — In the last example, the topology induced by T on E_K is identical with T_K, since by definition it is coarser than T_K and, since T is finer than Tᵤ, the topology induced by T o… 07EI
Definition 2 Let E be the vector space which is the direct sum (A, II, § 1.6) of the family of locally convex spaces (Eᵢ)_i ∈ I. For each i ∈ I, let fᵢ be the canonical injection of Eᵢ in E. By… 07EJ
Example — Let (aᵢ)_i ∈ I be a basis of the vector space E and consider the canonical topology (I, p. 2, Example 5) on each line Raᵢ; the direct sum of these topologies is the finest locall… 07EK
Proposition 6 A locally convex topology T on E is the direct sum of the topologies of the Eᵢ, if and only if, the following property holds: a linear mapping of E in a locally convex space G (res… 07EL
Corollary With the notation of prop. 5, II, p. 27, suppose that E is the sum of the g_α(F_α). Let F be the topological direct sum of the family (F_α)_α ∈ A, and let j_α : F_α → F be the cano… 07EM
Proposition 7 The canonical injection j : E → ∏_i ∈ I Eᵢ is continuous when E carries the direct sum topology of the Eᵢ and ∏_i ∈ I Eᵢ carries the product topology. When I is finite, this mappin… 07EN
Proposition 8 Let N_ι be a subspace of E_ι, for every ι ∈ I, 07EO
Corollary 1 If N_ι is closed in E_ι for every ι ∈ I, then N = ∑_ι N_ι in closed in E. 07EP
Corollary 2 If each E_ι is Hausdorff, so also is E and each E_ι is closed in E. 07EQ
Lemma 1 — Let 𝔉 be a Cauchy filter on E (for T); then there exists an integer k, such that for all N ∈ 𝔉 and every neighbourhood V of 0 in E, the subspace Eₖ meets N + V. 07ER
Proposition 9 Suppose that the topology induced on Eₙ by Tₙ₊₁ is identical with Tₙ for every integer n. Then (i) The topology induced by T on Eₙ is identical with Tₙ for each n; if the Tₙ are Ha… 07ES
Lemma 2 Let V be a convex neighbourhood of 0 in M, a vector subspace of a locally convex space F. Then there exists a convex neighbourhood W of 0 in F such that W ∩ M = V. Further, if M is… 07ET
Remark — 1) Suppose that E is the union of an increasing directed, non-enumerable family of subspaces (E_α)_α ∈ I, each E_α having a locally convex topology T_α, such that, for E_α ⊂ E_β… 07EU
Proposition 10 Let E, F be two locally convex spaces. Suppose that : ↩ 1 07EV
Corollary With the same hypotheses on E and F as in prop. 10 and assuming that E is Hausdorff, then every continuous surjective mapping v of F in E is a strict morphism. 07EW
Proposition 11 Let E be a metrisable locally convex space. The topology of E can be defined by a distance that is invariant under translations, and for which the open balls are convex. 07EX
Proposition 12 Let E and F be two Fréchet spaces and u a continuous linear mapping of E on F. Then there exists a section of u that is continuous though not necessarily linear. 07EY
Corollary If L is a compact set in F, then there exists a compact set K in E such that u(K) = L. 07EZ
Remark — 1) The corollary to prop. 12 can also be deduced from th. 1 of I, p. 17 and prop. 18 of GT, IX, § 2.10. 2) We keep the notations of prop. 12. Let p be a continuous semi-norm on E… 07F0

§ 5. Separation of convex sets

Theorem 1 (Hahn-Banach) — Let A be an open convex non empty set of the topological vector space E and let M be a non-empty linear variety which does not meet A. Then there exists a closed hyperplane H whi… 07FI
Remark 1 When 0 ∈ M, th. 1 can be stated as follows : there exists a continuous linear form in E, such that g(x) = 0 in M and g(x) > 0 in A (II, p. 8, prop. 4). 07FJ
Remark 2 If we apply theorem 1 to the case where E carries the finest locally convex topology (II, p. 25, Example 2), and if, for the sake of simplicity, we suppose that 0 ∈ A, then we get… 07FK
Definition 1 — Two non-empty sets A, B of a real topological vector space E are said to be separated by a closed hyperplane H if A is contained in one of the closed half-spaces determined by H… 07FL
Definition 2 — Two non-empty sets A, B of a real topological vector space are said to be strictly separated by the closed hyperplane H if A is contained in one of the open half-spaces determine… 07FM
Proposition 1 — Let A be an open non-empty convex set and let B be a non-empty convex set in a real topological vector space E; if A does not meet B then there exists a closed hyperplane that se… 07FN
Remark — 1) The hyperplane H does not meet A (II, p. 15, prop. 1; if A and B are two convex non-empty open sets that do not meet then there exists a closed hyperplane that separates A str… 07FO
Definition 3 — For a subset A of a vector space E, a hyperplane H is called a support-hyperplane of A, if H contains at least one point of A and all the points of A lie on the same side of H. 07FP
Proposition 2 — Let A be a non-empty compact subset of a topological vector space E. For every closed hyperplane H in E, there exists a support hyperplane of A parallel to H. 07FQ
Proposition 3 — In a topological vector space E, let A be a closed convex set with a non-empty interior. Then every support hyperplane of A is closed and every frontier point of A belongs to at… 07FR
Proposition 4 — Let A be a closed non-empty convex set in a locally convex space E and let K be a compact non-empty convex set in E, that does not meet A. Then there exists a closed hyperplane H… 07FS
Remark In a Hausdorff locally convex space E, let A and B be two closed non-empty convex sets that are disjoint, if E is finite dimensional then there exists a closed hyperplane that sepa… 07FT
Corollary 1 — In a locally convex space, every closed convex set A is the intersection of the closed half-spaces which contain it. 07FU
Corollary 2 — In a Hausdorff locally convex space, every compact convex set A is the intersection of the closed half-spaces which contain it and which are determined by support hyperplanes of… 07FV
Corollary 3 — In a locally convex space, the closure of each linear variety M is the intersection of the closed hyperplanes that contain M. 07FW
Corollary 4 Let C be a closed convex set in a locally convex space E. A subset A of E is contained in C, if, and only if, for every real valued continuous affine function u in E such that u(x)… 07FX
Corollary 5 In a locally convex space E, the closure of each convex cone C of vertex 0 is the intersection of closed half-spaces containing C determined by closed hyperplanes that pass through… 07FY
Lemma 1 — If a cone A, with vertex 0, is contained in an open half-space determined by a hyperplane H, then it is contained in a closed half-space determined by a hyperplane H₀, that is pa… 07FZ
Proposition 5 Let X be a closed convex set in a locally convex space E. Then every lower semi-continuous convex function f defined in X is the upper envelope of a family of functions that are th… ↩ 1 07G0
Remark 1 It follows from prop. 5 that f is the upper envelope of a directed increasing family of functions that are the restrictions to X of functions which are continuous and convex in E. 07G1
Remark 2 Suppose further that X is a closed convex cone with vertex 0 and that f is positively homogeneous. Then f is the upper envelope of a family of functions which are the restrictions… 07G2
Remark 3 The restriction to X of a continuous affine function in E is a function that is affine in X (i.e. both concave and convex II, p. 17); but it may be the case that there exist contin… 07G3
Proposition 6 Let f be an upper semi-continuous affine function in a compact convex set X, of the Hausdorff locally convex space E. Let L be the set of restrictions to X of continuous affine fun… 07G4
Corollary Let f be a continuous affine function in X; then there exists a sequence (hₙ) of elements of L which converges uniformly to f in X. 07G5

§ 6. Weak topologies

Proposition 1 Let F, G be two real vector spaces and B a bilinear form on F × G. Let 07H0
Example — 1) Let E be a vector space and let E be its dual. The canonical bilinear form (x, x^*) ↦ ⟨ x, x^* ⟩ on E × E^* (A, II, § 2.3) puts E and E in separating duality : for (D{II}) is… 07H1
Definition 1 — By the dual of a topological vector space E, we mean the subspace E' of E, the dual of the vector space E, formed by the continuous linear forms on E. 07H2
Remark — 1) When E is a topological vector space, the dual E^* of the vector space E will be called the algebraic dual of E to avoid confusion. We note also that E^* is the dual of the to… 07H3
Definition 2 — Let F and G be two vector spaces put in duality by the bilinear form B. The coarsest topology on F that makes all the linear forms B(., y): x ↦ B(x, y) continuous, where y varies… 07H4
Proposition 2 — The weak topology σ(F, G) is Hausdorff if and only if the duality between F and G is separating in F. 07H5
Proposition 3 — Let F and G be two real vector spaces in duality. Every linear form on F, that is continuous for σ(F, G), can be written as x ↦ ⟨ x, y ⟩ for some y ∈ G. The element y ∈ G is uniq… 07H6
Corollary 1 — A family (aᵢ) of points of F is total for the topology σ(F, G) if, and only if, for every y ≠ 0 in G, there exists an index i such that ⟨ aᵢ, y ⟩ ≠ 0. 07H7
Corollary 2 — A family (aᵢ) of points of F is topologically independent for the topology σ(F, G), if, and only if, for every index i, there exists an element bᵢ ∈ G such that : ⟨ aᵢ, bᵢ ⟩ ≠ 0… 07H8
Corollary 3 — Let G₁ and G₂ be two vector subspaces of F^*, in duality with F (for the restriction of the canonical bilinear form). Then σ(F, G₂) is finer than σ(F, G₁) if and only if G₁ ⊂ G₂. 07H9
Corollary 4 — Let G be a vector subspace of the dual F^*, of the vector space F. Then F and G are in separating duality (for the canonical bilinear form) if, and only if, G is dense in F^* in… 07HA
Definition 2 Let F and G be two (real) vector spaces in duality. For every set M of F we call the polar of M, the set of those y ∈ G for which ⟨ x, y ⟩ ≥ -1 for all x ∈ M. (For complex vector s… 07HB
Proposition 4 (i) For any set M of F, the polar set M^∘ is a convex set that contains 0 and is closed in G for the topology σ(G, F). 07HC
Theorem 1 (The bipolar theorem) Let F, G be two real vector spaces in duality. For every subset M of F the polar set M^∘ ∘ in F of the polar set M^∘ of M in G is the closed convex envelope (for σ(F, G)) of M ∪ 0. 07HD
Corollary 1 For any family (M_α) of closed convex sets of F (in the topology σ(F, G)), each containing 0, the polar set of the intersections M = ⋂ limits_α M_α is the convex closed envelope (f… 07HE
Corollary 2 For every vector subspace M of F, the subspace M^∘∘ is the closure of M in the topology σ(F, G). 07HF
Remark — Every neighbourhood of 0 in G in the topology σ(G, F) contains a neighbourhood V defined by a finite number of inequalities of the form |⟨ xᵢ, y ⟩| ≤ 1 (1 ≤ i ≤ n), where the xᵢ… 07HG
Corollary 3 Let T be the topology of a locally convex space E and let E' be its dual (II, p. 42, def. 1). 07HH
Proposition 5 Let u be a linear mapping of F in F₁. The following properties are equivalent : 07HI
Remark — Suppose that the duality between F and G is separating in G and that the duality between F₁ and G₁ is separating in G₁. If we identify G and G₁ with subspaces of F^* and F₁^* res… 07HJ
Corollary Suppose that the duality between F and G is separating in G. If u is a linear mapping of F in F₁, that is continuous for σ(F, G) and σ(F₁, G₁), then its transpose is a linear mappi… 07HK
Proposition 6 Suppose that the duality between F and G (resp. F₁ and G₁) is separating in G (resp. F₁). Let u be a linear mapping of F in F₁ that is continuous for σ(F, G) and σ(F₁, G₁). Let A b… 07HL
Corollary 1 With the notations of prop. 6, the relation u(A) ⊂ A₁ implies ᵗ u(A₁^∘) ⊂ A^∘; if further A₁ is convex, closed (for σ(F₁, G₁)) and contains the origin, then these two relations are… 07HM
Corollary 2 Let u be a linear mapping of F in F₁ that is continuous for σ(F, G) and σ(F₁, G₁). We have then 07HN
Proposition 7 — Let M be a vector subspace of F and N a vector subspace of G where F and G are two vector spaces in duality. Suppose that M and N are orthogonal (which is equivalent to saying th… 07HO
Remark — The duality between M and G/N (where M and N are two orthogonal subspaces) is separating in M, if and only if M ∩ G^∘ = 0; it is separating in G/N, if and only if N = M^∘. 07HP
Corollary 1 — Suppose that the duality between F and G is separating in F. For a vector subspace M of F the topology σ(G/M^∘, M) is identical with quotient topology of σ(G, F) by M^∘, if and o… 07HQ
Corollary 2 — If M is of finite dimension n and the duality is separating in F, then M^∘ is of codimension n in G. If M is closed for σ(F, G) and of finite codimension n and if the duality is… 07HR
Corollary 3 — Let (F, G), (F1, G1) be two pairs of spaces in separating duality and let u be a linear mapping of F in F1, which is continuous for σ(F, G) and σ(F₁, G₁). Then u is a strict morp… 07HS
Corollary 4 — Let (F, G), (F1, G1) be two pairs in separating duality, and let u be a linear mapping of F in F1 that is continuous for σ(F, G) and σ(F₁, G₁). Then u is surjective, if and only… 07HT
Remark — 1) Let (F₁, G₁), (F₂, G₂), (F₃, G₃) be three pairs of spaces in separating duality and consider a sequence of two linear mappings 07HU
Proposition 8 — Let (Fᵢ, Gᵢ)_i ∈ I be a family of pairs of spaces in duality. Let F = ∏_i ∈ I Fᵢ be the product space of the Fᵢ and G = ⨁_i ∈ I Gᵢ be the direct sum of the Gᵢ. If, for all x = (x… 07HV
Remark — The duality between F and G is separating in F (resp. in G) if and only if for all i ∈ I, the duality between Fᵢ and Gᵢ is separating in Fᵢ (resp. in Gᵢ). If the duality between… 07HW
Corollary 1 — Let F and G be two vector spaces in separating duality. If the space F (with σ(F, G)) is the direct topological sum of two subspaces M, N then the space G (with σ(G, F)) is the d… 07HX
Corollary 2 — Let (eᵢ)_i ∈ J be a basis of the vector space F with dual F^*, and let u : R⁽ᴵ⁾ → F be an (algebraic) isomorphism defined by this basis. Then the transposed mapping ᵗ u : F^* → R… 07HY
Proposition 9 — Let F, G be two vector spaces in separating duality. If hatF is the completion of the space F for the topology σ(F, G) and if we consider the canonical injection j : F → G^*, whe… 07HZ
Proposition 10 Let G be a real vector space and F = G^* its dual with the topology σ(G^*, G). 07I0
Lemma 1 Let E be a Hausdorff weak space and C a proper cone with vertex 0 in E, that is complete for the uniform structure induced by that of E. Every continuous linear form in E is then t… 07I1
Proposition 11 Let E be a Hausdorff weak space, and C be a proper convex cone with vertex 0 in E and which is complete in the uniform structure induced by that of E. 07I2
Corollary 1 — Let E be a Hausdorff weak space and C a proper convex cone of vertex 0 in E that is complete for the uniform structure induced by that of E. Then the mapping (x, y) ↦ x + y of C… 07I3
Corollary 2 — Let E be a Hausdorff weak space, and C a proper convex cone with vertex 0 in E, that is complete for the uniform structure induced by that of E. 07I4

§ 7. Extremal points and extremal generators

Definition 1 Let A be a convex set in an affine space E. Then we say that a point x ∈ A is an extremal point of A if there does not exist an open segment that is contained in A and contains x. 07IP
Example 1 In the space Rⁿ, all the points of the sphere Sₙ₋₁ are extremal points of the closed ball Bₙ. For, if ∑ᵢ yᵢ² ≤ 1,\ ∑ᵢ zᵢ² ≤ 1 and 0 < λ < 1, the relation λ² ∑ᵢ yᵢ² + (1 - λ)² ∑ᵢ zᵢ… 07IQ
Example 2 In the normed space B(N) of bounded sequences of real numbers (I, p. 4) the extremal points of the unit ball are the points x = (ξₙ) such that |ξₙ| = 1 for all n. For, suppose that… 07IR
Example 3 Let u : E → E' be an affine mapping of an affine space E in an affine space E'; let C ⊂ E,\ C' ⊂ E' be two convex sets such that u(C) ⊂ C'. If x' is an extremal point of C' and x i… 07IS
Proposition 1 Let B be the set of extremal points of A, a non-empty compact convex set in a Hausdorff locally convex space E, and let f be a convex function defined in A and upper semi-continuou… 07IT
Corollary Let A be a compact convex set in a Hausdorff locally convex space E. Then every closed support hyperplane H of A contains at least one extremal point of A. 07IU
Theorem 1 (Krein-Milman) — In a Hausdorff locally convex space E, every compact convex set A is the closed convex envelope of the set of its extremal points. 07IV
Proposition 2 Let x be an extremal point of a compact convex set A in a Hausdorff locally convex space E. Then for every open neighbourhood V of x in E, there exists an open half-space F in E su… 07IW
Corollary — In a Hausdorff locally convex space let K be a compact subset of a compact convex set A. Then the following conditions are equivalent. 07IX
Remark 1 Even if the vector space E is finite dimensional the set of extremal points of a compact convex set is not necessarily closed (II, p. 89, exerc. 11). 07IY
Remark 2 If K is a compact set in a non complete Hausdorff locally convex space, and A, the closed convex envelope of K is not compact, there can be extremal points of A that do not belong… 07IZ
Remark 3 In a Banach space E of infinite dimension, it may happen that the closed ball of centre 0 and radius 1 does not possess any extremal point (II, p. 89, exerc. 14). 07J0
Remark 4 If A is a compact convex set in a Hausdorff locally convex space, it may happen that an extremal point of A does not belong to any support hyperplane of A (II, p. 78, exerc. 11). T… 07J1
Definition 2 Let C be a convex cone of vertex 0 in a vector space E. We say that a half-line D ⊂ C originating at 0 is an extremal generator of C, if every open segment contained in C, not cont… 07J2
Remark 1 Let C be a pointed proper convex cone in E, and consider on E the order structure for which C is the set of elements ≥ 0 (II, p. 12, prop. 13); in order that an element of E, say x… 07J3
Proposition 3 In a vector space E, let C be a convex cone with vertex 0, and let x₀ ≠ 0 be a point of C, and D a half-line that is contained in C, originating from 0 and containing x₀. Let H be… 07J4
Definition 3 Let C be a convex set in a Hausdorff topological vector space E. A compact convex non-empty set A of C is called a cap of C if the complement C − A of A in C is convex. 07J5
Proposition 4 — Let C be a pointed convex cone with vertex 0 in E, a Hausdorff locally convex space. 07J6
Corollary 1 — Let C be a pointed convex cone of vertex 0 in E, a Hausdorff locally convex space and let p be the gauge of A, a cap of C. The extremal points of A are then the point 0, and the… 07J7
Corollary 2 — Every point of C that belongs to a cap of C, also belongs to the convex closed envelope of the union of the extremal generators of C. 07J8
Proposition 5 Let C be a proper convex cone with vertex 0 in E, a Hausdorff weak space; suppose that C is complete for the uniform structure induced by that of E, and that there is an enumerable… 07J9
Remark 2 There exist proper convex cones that are weakly complete and which have no extremal generator (II, p. 92, exerc. 31). 07JA
Proposition 6 Let E be a Hausdorff locally convex space and K a convex compact set in E which does not contain 0. Then the smallest pointed cone C of vertex 0 which contains K is a proper convex… 07JB
Example 1 Every proper closed convex cone in E, a vector space of finite dimension, has a compact sole. In fact, by II, p. 52, prop. 11 we need only consider the case where E = Rⁿ and C = R₊… 07JC
Example 2 \ If X is a compact space, then the cone M₊(X) of positive measures on X, with the vague topology, is a cone with a compact sole (INT, III, 2nd ed., § 1, No. 9, cor. 3 of prop. 15)… 07JD

§ 8. Complex locally convex spaces

Remark — Given a topological vector space F over R, it is not always the case that there exists an automorphism u of F whose square is -1_F; for example, it is not possible to define vect… 07KJ
Theorem 1 (Hahn-Banach) — Let V be a vector subspace of E, a complex vector space, and let f be a (complex) linear form on V and p a semi-norm on E such that |f(y)| ≤ p(y) for all y ∈ V. Then there exists… 07KK
Corollary 1 — Let x₀ be a point of a complex topological vector space E and p be a continuous semi-norm in E; then there exists a continuous (complex) linear form f defined in E, such that f(x… 07KL
Corollary 2 — Let V be a vector subspace of a complex locally convex space E and f be a (complex) linear form defined and continuous in V; then there exists a continuous linear form f₁ defined… 07KM
Corollary 3 — Let M be a finite dimensional vector subspace of a Hausdorff complex locally convex space E. Then there exists a closed vector subspace N of E that is a topological complement of… 07KN
Proposition 1 — Let A be an open non-empty convex set in a complex topological vector space E and M be a non-empty (complex) linear variety that does not meet A. Then there exists a closed compl… 07KO
Corollary — In a complex locally convex space E, every closed complex linear variety M is the intersection of the closed complex hyperplanes which contain it. 07KP
Proposition 2 Let A be a non-empty balanced open convex set of a complex topological vector space E, and B be a non-empty convex set that does not meet A. Then there exists a continuous complex… 07KQ
Proposition 3 Let A be a balanced, closed, convex set in a complex locally convex space E and let K be a non-empty compact convex set in E that does not meet A. Then there exists a continuous co… 07KR
Definition 1 Let F and G be two complex vector spaces in duality. For any subset M of F, the polar of M in G, denoted by M^∘, is the set of y ∈ G such that Re(⟨ x, y ⟩) ≥ -1 for all x ∈ M. 07KS

III. Spaces of continuous linear mappings

§ 1. Bornology in a topological vector space

Definition 1 — A bornology on a set E is a subset B of the set of all subsets of E satisfying the following conditions (cf. GT, X, § 1.2, Remark 2). (B1) Every subset of a set of B belongs to B… 07KW
Example — Let E be a metric space; the set of all bounded subsets of E (GT, IX, § 2, No. 2) is a covering bornology on E. Let G be the group of isometries of E; the set of all subsets M of… 07KX
Definition 2 Let E be a vector space. A bornology B on E is said to be convex, if for every X ∈ B and t ∈ K, the homothetic set tX and the convex balanced envelope Γ(X) (II, p. 10) of X belong… 07KY
Definition 3 Let E be a topological vector space. A subset A of E is said to be bounded if it is absorbed by every neighbourhood of 0 in E (I, p. 7, def. 4). 07KZ
Remark — 1) If E is a semi-normed space, the balls form a fundamental system of bounded neighbourhoods of 0 in E. Conversely, if E is a locally convex topological vector space, and if the… 07L0
Definition 4 Let E be a locally convex space. A bornology B on E is said to be adapted to E, if it is convex, is composed of bounded subsets of E and if the closure of every set of B belongs to… 07L1
Proposition 1 Let E be a locally convex space. The set of bounded subsets of E is an adapted bornology. 07L2
Definition 5 Let E be a locally convex space. The set of all bounded subsets of E is called the canonical bornology of E. 07L3
Proposition 2 In a locally convex space E, every precompact set is bounded. Let A be a precompact subset of E, and V be a convex balanced neighbourhood of 0. There exists a finite sequence (aᵢ)_… 07L4
Corollary In a locally convex space, the set of points of a Cauchy sequence is bounded. In fact, this set is precompact (GT, II, § 4, No. 2). 07L5
Remark 5 — In general the bounded subsets of a locally convex space E are not all precompact (for example, if E is an infinite dimensional normed space, its unit ball is not compact (I, p… 07L6
Proposition 3 — Let A be a subset of a locally convex space E. Suppose that A is bounded; then for every sequence (xn) of points of A and for every sequence (\lambdan) of scalars tending to 0, t… 07L7
Corollary — A subset A of E is bounded if and only if every countable subset of A is bounded. 07L8
Proposition 4 — Let E and F be two locally convex spaces and f : E \to F a continuous mapping. Assume that f(0) = 0 and that there exists a real number m \geq 0 such that f(\lambda x) = \lambda^… 07L9
Corollary 1 — Let E and F be two locally convex spaces, and u : E \to F be a continuous linear mapping. If A is a bounded subset of E, then u(A) is bounded in F. 07LA
Corollary 2 — Let E = \prod{i \in I} Ei be the product of a family of locally convex spaces. Then a subset of E is bounded if and only if all its projections are bounded. 07LB
Corollary 3 — Let E be a vector space, (Fi){i \in I} a family of locally convex spaces and fi a linear mapping from E into Fi (for i \in I). Suppose E is assigned the coarsest topology (locall… 07LC
Corollary 4 Let Eᵢ (1 ≤ i ≤ n) and F be locally convex spaces, and f be a continuous multilinear map from ∏ᵢ₌₁ⁿ Eᵢ into F. If Bᵢ is a bounded subset of Eᵢ, for 1 ≤ i ≤ n, then f(∏ᵢ₌₁ⁿ Bᵢ) is b… 07LD
Corollary 5 Let E and F be two locally convex spaces and u : E → F be a continuous polynomial mapping. If A is a bounded subset of E, then u(A) is bounded. 07LE
Proposition 5 Let (Eᵢ)_i ∈ I be a family of Hausdorff locally convex spaces, and let E be the topological direct sum of this family (II, p. 29). In order that a subset B of E be bounded, it is n… 07LF
Proposition 6 Let E be a locally convex space which is the strict inductive limit of an increasing sequence (Eₙ) of closed vector subspaces of E (II, p. 33). A subset B of E is bounded if and on… 07LG
Proposition 7 — Let (Eₙ)_n ≥ 0 be a sequence of Hausdorff locally convex spaces, and for every n, let uₙ : Eₙ → Eₙ₊₁ be an injective linear mapping which is compact (i.e. such that there exists… 07LH
Lemma 1 Under the hypothesis of prop. 7, the topology of E is the finest topology for which all the mappings vₙ : Eₙ → E are continuous. 07LI
Remark 1 With the above notations, let Fₙ be the vector space generated by Hₙ, with a norm equal to the gauge of Hₙ. We shall see (III, p. 8, cor.) that Fₙ is a Banach space. The injection… 07LJ
Remark 2 Let F be a locally convex space, k an integer ≥ 0 and f : Eᵏ → F a multilinear mapping. For f to be continuous, it is necessary and sufficient that the restriction of f to Eₙᵏ is c… ↩ 2 07LK
Proposition 8 Let E be a Hausdorff locally convex space and let A be a closed, balanced, bounded and convex subset of E. Let (xₙ) be a Cauchy sequence in E_A. Then this sequence converges in E_A… 07LL
Corollary — If A is semi-complete (in particular, complete) then EA is a Banach space. In fact, a Cauchy sequence in EA is also a Cauchy sequence for the topology of E and is contained in a… 07LM
Definition 6 — A locally convex space E is said to be quasi-complete if every closed and bounded subset of E is complete (for the uniform structure induced by that of E). A complete locally con… 07LN
Proposition 9 — (i) A closed vector subspace of a quasi-complete locally convex space is quasi-complete. (ii) The product of quasi-complete locally convex spaces is quasi-complete. (iii) The top… 07LO
Proposition 10 — Let E be a locally convex space, M a vector subspace of E such that every point of E is in the closure of a bounded subset of M. Then every continuous linear mapping f from M int… 07LP

§ 2. Bornological spaces

Lemma 1 — Let G be a semi-normed space, p its semi-norm, and u a linear mapping from G into E. The following conditions are equivalent : 07M8
Proposition 1 The following conditions are equivalent : (i) Every semi-norm on E which is bounded on bounded subsets of E is continuous. (i') Every convex balanced subset of E which absorbs the… 07M9
Definition 1 A locally convex space is said to be bornological if it satisfies the equivalent conditions of prop. 1. 07MA
Example — 1) Every semi-normed space is bornological. 2) In particular, every finite dimensional locally convex space is bornological. 3) On account of the transitivity of final locally co… 07MB
Corollary Every Hausdorff and semi-complete bornological space is an inductive limit of Banach spaces. 07MC
Proposition 2 A locally convex metrizable space is bornological. 07MD
Corollary — Every Fréchet space (II, p. 24) is the inductive limit of Banach spaces. 07ME

§ 3. Spaces of continuous linear mappings

Remark — 1) The 𝔖-topology is identical with the 𝔖'-topology, where 𝔖' denotes the bornology generated by 𝔖 (III, p. 1). 07MJ
Proposition 1 Let E be a set, 𝔖 a family of subsets of E, F a topological vector space and H a vector subspace of Fᴱ. In order that the 𝔖-topology be compatible with the vector space structure o… 07MK
Corollary Let E and F be two locally convex spaces, 𝔖 a family of bounded subsets of E, and L(E; F) the vector space of continuous linear mappings from E into F. Then the 𝔖-topology is compa… 07ML
Example 1 If 𝔖 is the set of all finite subsets of E, then the 𝔖-topology is the topology of simple convergence and the space L_𝔖(E; F) is also denoted by Lₛ(E; F). A bounded subset of Lₛ(E… 07MM
Example 2 If 𝔖 is the set of compact (resp. precompact, compact convex) subsets, then the 𝔖-topology is called the topology of compact (resp. precompact, compact convex) convergence and the… 07MN
Example 3 If 𝔖 is the set of all bounded subsets of E, we say that the 𝔖-topology is the topology of bounded convergence and the space L_𝔖(E; F) is denoted by L_b(E; F). 07MO
Example 4 When F = K, the space L(E; F) is the dual E' of E. We denote by E'_𝔖, E'ₛ etc. the space L_𝔖(E; K), Lₛ(E; K) etc. The space E'ₛ (resp. E'_b) is called the weak dual (resp. strong d… ↩ 1 07MP
Remark — 3) Let A be a dense subset of the unit ball of E. In view of the continuity of u, we also have ↩ 1 07MQ
Proposition 2 — Let E and F be two locally convex spaces and let 𝔖 be a set of bounded subsets of E. 07MR
Proposition 3 — Let E and F be two locally convex spaces, F being assumed Hausdorff, and let 𝔖 be a family of bounded subsets of E. If the union A of the sets of 𝔖 is total in E, then the space… 07MS
Proposition 4 Let E, F be two locally convex spaces, and F be Hausdorff. Let the space Fᴱ of all mappings from E into F be assigned the topology of simple convergence. Then 07MT
Corollary 1 — For an equicontinuous subset H of L(E; F) to be relatively compact in Lₛ(E; F), it is necessary and sufficient that for all x ∈ E, the set H(x) of all u(x) as u ranges over H, is… 07MU
Corollary 2 — Every equicontinuous subset of the dual E' of E is relatively compact for the weak topology σ(E', E) on E' (III, p. 14, Example 4). 07MV
Corollary 3 — In the strong dual E'_b of a semi-normed space E, every closed ball is compact for the weak topology σ(E', E). 07MW
Proposition 5 — Let E and F be two locally convex spaces and let T be a total subset of E. The following uniform structures coincide on every equicontinuous subset H of L(E; F): 07MX
Example — 1) Let μ be the Lebesgue measure on R, and let E be the semi-normed space Lᵖ(μ) (1 ≤ p < ∞) (INT, IV). For every numerical function f and every real number h, let fₕ be the funct… 07MY
Corollary — Suppose F is Hausdorff. Let H be an equicontinuous subset of L(E; F). If a filter Φ on H converges simply to a mapping u₀ from E into F, then u₀ is a continuous linear mapping fr… 07MZ
Proposition 6 — Let H be an equicontinuous subset of L(E; F). If F is metrizable and if there exists a countable total set in E, then the uniform structure on H of simple convergence in E is met… 07N0
Corollary 1 — Let E be a locally convex metrizable space, and F a normed space. Suppose that E and F both satisfy the first axiom of countability. Then L(E; F) is the union of a countable fami… 07N1
Corollary 2 — Every closed ball in the dual E' of a normed space satisfying the first axiom of countability, is a compact metrizable space for the weak topology σ(E', E), and for this topology… 07N2
Proposition 7 — Let M be a subset of E'. The following conditions are equivalent : 07N3
Corollary 1 — The topology of E is identical with the topology of uniform convergence on equicontinuous subsets of E'. 07N4
Corollary 2 — Let S be a family of bounded subsets of E. The S-topology on L(E; F) is the topology of uniform convergence on sets of the form A × B ⊂ E × F', where A is in S, and B belongs to… 07N5
Proposition 8 — Let H be a family of linear mappings from E into a locally convex space F. For H to be equicontinuous, it is necessary and sufficient that for every equicontinuous subset X in th… 07N6
Theorem 1 (Grothendieck) — Let E be a locally convex space, and let \mathscr{S} be an adapted and covering bornology on E. Let F \subset E^ be the space of those linear forms on E whose restriction to each… 07N7
Lemma 1 Let A be a closed convex balanced subset of E and let u be a linear form on E whose restriction to A is continuous. Then for every \varepsilon 0, there exists an x' \in E' such tha… 07N8
Theorem 2 Let E be a locally convex Hausdorff space and hatE its completion; we identify E and hatE with vector subspaces of E'^*. Then for an element f ∈ E'^* to belong to hatE, it is neces… 07N9
Corollary 1 (Banach) — Let E be a Hausdorff and complete locally convex space. In order that a linear form on E' be continuous for the weak topology σ(E', E) (i.e. arises from an element of E) it is su… 07NA
Remark — Suppose in addition, that there exists a countable total set in E; then every equicontinuous subset of E' is metrizable for the topology σ(E', E) (III, p. 18, prop. 6); therefore… 07NB
Corollary 2 Let (Eᵢ)_i ∈ I be a family of Hausdorff locally convex spaces and let E be their topological direct sum. Then the canonical mapping from the direct sum of the hatEᵢ into hatE is an… 07NC
Proposition 9 Let E and F be two locally convex spaces and S a family of bounded subsets of E. Then every equicontinuous subset of L(E; F) is bounded for the S-topology. 07ND
Proposition 10 Let E be a bornological space (in particular, a metrizable locally convex space) and F a locally convex space. Every subset H of L(E; F) which is bounded for the topology of bounde… 07NE
Proposition 11 Let E and F be two locally convex spaces, S a cover of E consisting of bounded subsets. If F is Hausdorff and quasi-complete (III, p. 8), then every equicontinuous subset H of L(E… 07NF
Remark 1 — Let M be a complete uniform subspace of L_S(E; F). For every set of bounded subsets S' ⊃ S of E, the S'-topology is finer than the S-topology on L(E; F); on the other hand, there… 07NG
Corollary Let E and F be two locally convex spaces, H an equicontinuous subset of L(E; F). If F is Hausdorff and quasi-complete and if a filter Φ on H converges simply at all points of a tot… 07NH
Remark 2 Let (uₙ) be a sequence of continuous linear mappings from a Banach space E into a Banach space F; it may happen that (uₙ(x)) has a limit at every point of an everywhere dense vecto… 07NI
Proposition 12 — Let E be a bornological locally convex space, F a complete locally convex Hausdorff space and 𝔖 a family of bounded subsets of E containing the image of every sequence converging… 07NJ
Corollary 1 — The strong dual of a bornological space is complete. 07NK
Corollary 2 — Let E be a semi-normed space, and F a Banach (resp. Fréchet) space. The space L_b(E; F) is a Banach (resp. Fréchet) space. In particular, the dual of a semi-normed space is a Ban… 07NL

§ 4. The Banach-Steinhaus theorem

Proposition 1 Let T be a subset of E. The following conditions are equivalent: 07NZ
Corollary The following conditions are equivalent: 07O0
Definition 1 A set T satisfying the equivalent conditions of prop. 1 is said to be a barrel in E. 07O1
Definition 2 A space E is said to be barrelled if it satisfies the equivalent conditions of the corollary of prop. 1. 07O2
Scholium In the dual of a barrelled space, the class of equicontinuous sets, the class of strongly bounded sets, the class of weakly bounded sets and the class of relatively compact sets fo… 07O3
Proposition 2 Every locally convex space E which is a Baire space (GT, IX, § 5, No. 3) is barrelled. 07O4
Corollary — Every Fréchet space (and in particular, every Banach space) is barrelled. This follows from Baire’s theorem (GT, IX, § 5, No. 3, th. 1). 07O5
Proposition 3 — Let (Fᵢ)_i ∈ I be a family of barrelled spaces, and for every i ∈ I, let fᵢ be a linear mapping from Fᵢ into a vector space E. The space E with the finest locally convex topology… 07O6
Corollary 1 — Every quotient space of a barrelled space is barrelled. 07O7
Corollary 2 — Let (Eᵢ)_i ∈ I be a family of locally convex spaces and let E be the topological direct sum of this family. For E to be barrelled, it is necessary and sufficient that each Eᵢ is… 07O8
Corollary 3 — Every inductive limit of barrelled spaces is a barrelled space. 07O9
Theorem 1 — Let E be a barrelled space, F a locally convex space. Every simply bounded subset H of L(E; F) is equicontinuous. 07OA
Corollary 1 — Let E and F be two Banach spaces, H a family of continuous linear mapping from E into F; if, for all x ∈ E, we have sup_u ∈ H |u(x)| < +∞, we also have sup_u ∈ H |u| < +∞. 07OB
Corollary 2 — (Banach-Steinhaus theorem). — Let E be a barrelled space, F a locally convex Hausdorff space, and let (uₙ) be a sequence of continuous linear mappings from E into F, which conver… 07OC
Remark 1 The property expressed by cor. 2 does not imply that E is barrelled : we shall see later that the strong dual of a Fréchet space possesses this property, while not necessarily bein… 07OD
Remark 2 Let E and F be two Banach spaces, and (uₙ) a sequence of continuous linear mappings from E into F such that sup |uₙ| = + ∞. Then the set X of all x ∈ E such that sup |uₙ(x)| = + ∞… 07OE
Corollary 3 — Let E be a barrelled space, F a locally convex Hausdorff space and Φ a filter on L(E; F) which converges simply in E to a mapping u from E into F. If Φ contains a simply bounded… 07OF
Example — Let E be the Banach space (over C) consisting of continuous complex functions with period 1 in R, with the norm |f| = supₓ |f(x)|. 07OG
Corollary 4 — Let E and F be two locally convex spaces, 𝔖 a cover of E consisting of bounded subsets. If E is barrelled and F Hausdorff and quasi-complete, the space L_𝔖(E; F) is Hausdorff and… 07OH
Corollary 5 — The strong dual and the weak dual of a barrelled space are quasi-complete. 07OI
Theorem 2 — Let E be a locally convex Hausdorff space, F a locally convex space and 𝔖 a family of closed, convex, balanced, bounded and semi-complete subsets of E (III, p. 7). Every simply b… 07OJ
Corollary 1 — Let E be a locally convex Hausdorff space, F a locally convex space, and 𝔖 a family of bounded subsets of E. If E is semi-complete, then every simply bounded subset of L(E; F) is… 07OK
Corollary 2 — Every semi-complete bornological space is barrelled. 07OL
Corollary 3 — Let E be a locally convex space. Every subset of E which is bounded for σ(E, E') is bounded. 07OM

§ 5. Hypocontinuous bilinear mappings

Definition 1 A bilinear mapping u from E × F into G is said to be separately continuous if, for all x ∈ E, the linear mapping u(x, .) from F into G is continuous, and for all y ∈ F, the linear… 07P9
Proposition 1 For a bilinear mapping u from E × F into G to be separately continuous, it is necessary and sufficient that for all y ∈ F, the linear mapping u(., y) from E into G is continuous an… 07PA
Proposition 2 Let E, F and G be three locally convex spaces. Suppose that E and F are metrizable and E is barrelled. Let H be a set of separately continuous bilinear mappings from E × F into G… 07PB
Corollary 1 — Let E and F be two Fréchet spaces, and G a locally convex space. Every separately continuous bilinear mapping from E × F into G is continuous. 07PC
Corollary 2 — Suppose that E and F are metrizable and that E is barrelled. Then φ is a topological vector space isomorphism from B(E, F) onto L_b(E; F'_b). 07PD
Proposition 3 — Let E, F, G be three locally convex spaces, 𝔖 a family of bounded subsets of E. Let u be a separately continuous bilinear mapping from E × F into G. The following properties are… 07PE
Definition 2 — Let u be a bilinear mapping from E × F into G. We say that u is 𝔖-hypocontinuous if u is separately continuous and if it verifies one of the equivalent conditions a), b), c) of p… 07PF
Proposition 4 Let u be a S-hypocontinuous bilinear mapping from E × F into G. For every set M ∈ S, the restriction of u to M × F is continuous, and u(M × Q) is bounded in G for every bounded sub… 07PG
Proposition 5 Let u be a (S, T)-hypocontinuous bilinear mapping from E × F into G. For every pair of sets M ∈ S, N ∈ T, u is uniformly continuous on M × N. 07PH
Proposition 6 If F is a barrelled space, every separately continuous bilinear mapping u from E × F into a locally convex space G is S-hypocontinuous for every set S of bounded subsets of E. 07PI
Remark — Suppose the topology of F is the finest locally convex topology on F for which the linear mappings h_α : F_α → F are continuous (II, p. 27). Then condition c) of prop. 3 (III, p… 07PJ
Proposition 7 Let E, F, G be three locally convex spaces, G being assumed Hausdorff; let E₀ (resp. F₀) be a dense vector subspace of E (resp. F). Let u be a separately continuous bilinear mappin… 07PK
Proposition 8 Let E, F, G be three locally convex spaces; assume that G is Hausdorff and quasi-complete. Let E₀ (resp. F₀) be a dense vector subspace of E (resp. F), 𝔖₀ (resp. 𝔗₀) a family of bo… 07PL
Proposition 9 Let R, S, T be three locally convex Hausdorff spaces. Suppose that the spaces L(R; S), L(S; T), L(R; T) are each assigned the topology of simple (resp. compact, bounded) convergenc… 07PM
Corollary 1 For every equicontinuous subset H of L(S; T) the map (u, v) ↦ v ∘ u from L(R; S) × H into L(R; T) is continuous. 07PN
Corollary 2 Suppose S is barrelled. If the sequence (uₙ) tends to u in L(R; S) and the sequence (vₙ) to v in L(S, T), then the sequence (vₙ ∘ uₙ) tends to v ∘ u in L(R; T). 07PO

§ 6. Borel's graph theorem

Theorem 1 — Let E be a locally convex space which is the inductive limit of Banach spaces, F a Souslin locally convex space, for example a Lusin space (GT, IX, § 6, No. 2 and No. 4), and u a… 07Q4
Remark Recall (III, p. 12) that every homological Hausdorff and semi-complete space, for example every Fréchet space, is the inductive limit of Banach spaces. \ This is also true for the… 07Q5
Proposition 1 — Let E be a Hausdorff locally convex space. Suppose that there exists a sequence (En){n \in \mathbf{N}} of Fréchet spaces satisfying the first axiom of countability, and continuou… ↩ 1 07Q6
Example 1 Every Fréchet space satisfying the first axiom of countability is a polish space, hence a Lusin space. Consequently, so are the spaces C(X), where X is locally compact and has a co… 07Q7
Theorem 2 — Let E be a locally convex space, which is the inductive limit of an increasing sequence (Eₙ)_n ∈ N of subspaces of E, endowed with the topologies of Fréchet spaces satisfying the… 07Q8
Corollary — The hypotheses on E being as in th. 2, assume, in addition that every bounded subset of E is relatively compact. Then the strong dual of E is a Lusin space. In particular, the st… 07Q9
Remark — Let E be as in th. 2; let F be a Hausdorff locally convex space which is the union of the images of a sequence of continuous linear mappings uₙ : Fₙ → F, where each Fₙ is a Fréch… 07QA
Proposition 2 — Let E be a Banach space, F a locally convex space and u a linear mapping from E into F. Assume that for every closed subset B of F, every compact subset X of E and every measure… 07QB

IV. Duality in topological vector spaces

§ 1. Duality

Definition 1 A locally convex topology T on E is said to be compatible with the duality between E and F if d_B is a bijection from F onto the dual of the locally convex space obtained by assign… 07QI
Proposition 1 (i) The closed convex subsets in E are the same for all the locally convex topologies on E which are compatible with the duality between E and F. (ii) The bounded subsets of E are… 07QJ
Definition 2 Let E and F be two spaces in duality. The Mackey topology on E, denoted by τ(E, F) is defined as the 𝔖-topology on E, where 𝔖 is the family of all subsets of F whose image in E^* (… 07QK
Remark 1 — Let A be a compact convex subset of a Hausdorff locally convex space G, and let tildeA be the closed convex balanced envelope of A. When the field K is R, the set tildeA is the c… 07QL
Theorem 1 (Mackey) — Let E and F be two spaces in duality; suppose that the duality is separating in F. In order that a locally convex topology T on E be compatible with the duality between E and F… 07QM
Lemma 1 The subspace F of E consists of all linear forms on E which are continuous for τ(E, F). 07QN
Lemma 2 Let T be a locally convex topology on E such that every linear form on E which is continuous for T belongs to F. Then T is coarser than τ(E, F). 07QO
Corollary Let p be a semi-norm on E. The following conditions are equivalent : 07QP
Remark 2 Let K be a convex subset of F which is compact for the weak topology σ(F, E) and μ a positive measure on K. Put 07QQ
Example Let G be a locally convex space and G' its dual. On G', the weak topology σ(G', G) and the topology of convex compact convergence (III, p. 14) are compatible with the duality betwe… 07QR
Definition 3 Let E and F be two vector spaces in duality, and T the family of subsets of F which are bounded for σ(F, E). Then the T-topology on F is denoted by β(E, F). 07QS
Proposition 2 Let E be a locally convex space, and let A be a convex subset of E (for example, a vector subspace of E). The closure of A is the same for the initial topology and for the weakened… 07QT
Remark 1 For a family (xᵢ)_i ∈ I of elements of E to be total (resp. topologically independent) for the initial topology, it is necessary and sufficient that it is so for the weakened topol… 07QU
Remark 2 Let T₁ and T₂ be two locally convex topologies on E, compatible with the duality between E and E', T₁ being finer than T₂. Then every neighbourhood of 0 for T₁, which is convex and… 07QV
Remark 3 Suppose E is Hausdorff (for the initial topology). Let A be a subset of E which is closed and bounded for σ(E, E'), hence also for every topology compatible with the duality betwee… ↩ 1 07QW
Proposition 3 Suppose E is Hausdorff, and E' its dual. Every subset of E which is precompact for the initial topology and compact for σ(E, E'), is compact for the initial topology. 07QX
Remark 4 The topology β(E, E') (IV, p. 4, def. 3) is finer than the Mackey topology. If β(E, E') is distinct from τ(E, E'), it is not compatible with the duality between E and E'. The space… 07QY
Proposition 4 Let E be a locally convex space. The Mackey topology on E is identical with the initial topology in each of the following cases : 07QZ
Proposition 5 Let u be a continuous linear mapping from E₁ into E₂. 07R0
Proposition 6 Let u be a linear mapping from E₁ into E₂ which is continuous for the weakened topologies. For i = 1, 2, let 𝔖ᵢ be a family of bounded subsets of Eᵢ. In order that ᵗ u is a continu… 07R1
Corollary Let u be a continuous linear mapping from E₁ into E₂. Then ᵗ u is continuous when the duals E'ᵢ are assigned the following topologies: a) the weak topologies σ(E'ᵢ, Eᵢ); b) the str… 07R2
Proposition 7 — Let u : E₁ → E₂ be a linear mapping. We assume that u is continuous for the weakened topologies of E₁ and E₂. 07R3
Corollary — Suppose E₁ is a normed space. Let u be a linear mapping from E₁ into E₂. The following properties are equivalent : 07R4
Proposition 8 — (i) Let E be a normed space, with dual E'. For every x ∈ E, we have 07R5
Remark 1 Formula (3) is a particular case of (4), corresponding to the linear mapping λ ↦ λ x from K into E. 07R6
Remark 2 Put B(x, y') = ⟨ u(x), y' ⟩ = ⟨ x, ᵗu(y') ⟩ for x ∈ E₁,\ y' ∈ E'₂. The above proof shows that B is a continuous bilinear form on E₁ × E'₂, with norm (GT, X, § 3, No. 2) equal to |u… 07R7
Corollary — Let dotE be a normed space satisfying the first axiom of countability. There exists a countable subset D of E' - 0 such that we have 07R8
Proposition 9 — (i) For a subset A of (E/M)' to be equicontinuous, it is necessary and sufficient that π(A) is an equicontinuous subset of E'. 07R9
Proposition 10 — (i) For a subset A of M' to be equicontinuous, it is necessary and sufficient that it is the image under ᵗ i of an equicontinuous subset of E'. 07RA
Remark — We know (II, p. 48, prop. 7, (ii)) that ι is a topological vector space isomorphism from E'ₛ/M^∘ onto M'ₛ (weak duals). For the topology of compact convex convergence, prop. 10 s… 07RB
Proposition 11 — (i) The weakened topology on E/M is the quotient of that on E; the weakened topology on M is induced by that of E. 07RC
Lemma 3 — (i) For every i ∈ I, let 𝔖ᵢ be a family of subsets of Fᵢ, which is bounded for σ(Fᵢ, Eᵢ); put 𝔖 = ⋃_i ∈ I 𝔖ᵢ. Then the 𝔖-topology on E is the product of the 𝔖ᵢ-topologies on the… 07RD
Proposition 12 The topology β(F, E) is the direct sum of the topologies β(Fᵢ, Eᵢ). The topology β(E, F) is the product of the topologies β(Eᵢ, Fᵢ). 07RE
Proposition 13 The Mackey topology τ(F, E) is the direct sum of the Mackey topologies τ(Fᵢ, Eᵢ). The topology τ(E, F) is the product of the topologies τ(Eᵢ, Fᵢ). 07RF
Proposition 14 (i) The mapping θ is an isomorphism from the strong (resp. weak) dual of S = ⨁_i ∈ I Eᵢ onto the product of the strong (resp. weak) duals of the Eᵢ: 07RG
Proposition 15 — (i) The map φ is an isomorphism from the topological direct sum of the strong duals of the Eᵢ onto the strong dual of P = ∏_i ∈ I Eᵢ. 07RH
Corollary — Every product of barrelled spaces is barrelled. 07RI

§ 2. Bidual. Reflexive spaces

Definition 1 — Let E be a locally convex space and E'_b its strong dual. The dual of the locally convex space E'_b is called the bidual of E and is denoted by E''. 07S8
Proposition 1 — The kernel of c_E : E → E'' is the closure of 0 in E. If E is Hausdorff, c_E is injective. 07S9
Proposition 2 — Suppose that the space E is bornological or barrelled. The initial topological on E is the inverse image under c_E of the strong topology on E''. 07SA
Proposition 3 — Let E be a locally convex Hausdorff space. In order that the strong dual E'_b of E be barrelled, it is necessary and sufficient that every subset of E'' which is bounded for σ(E'… 07SB
Remark Let E be a locally convex Hausdorff space, E' its dual and E'' its bidual. We have E ⊂ E'' ⊂ E'^*, where E'^* is the algebraic dual of E'. If B is a bounded subset of E, its closur… 07SC
Definition 2 — Let E be a locally convex space. We say that E is semi-reflexive if the canonical mapping c_E from E into E'' is bijective. 07SD
Theorem 1 — A locally convex Hausdorff space is semi-reflexive if and only if every bounded subset of E is relatively compact for the weakened topology σ(E, E'). If E is semi-reflexive, the… ↩ 1 07SE
Corollary — Let E be a locally convex semi-reflexive space. Every closed vector subspace M of E is semi-reflexive; moreover, the strong topology on E'/M^∘ (considered as the dual of M) is th… 07SF
Remark 1 Suppose E is semi-reflexive. Every subset of E which is convex, closed and bounded for the initial topology is compact for the topology σ(E, E') (IV, p. 1, prop. 1). On the other h… 07SG
Remark 2 By remark 3 of IV, p. 5, we can reformulate th. 1 as follows : the Hausdorff space E is semi-reflexive if and only if it is quasi-complete for its weakened topology. If it is semi-… 07SH
Remark 3 Under the hypotheses of the above corollary, the space E/M is not necessarily semi-reflexive (IV, p. 63, exerc. 10). 07SI
Definition 3 — A locally convex space E is said to be reflexive if the canonical mapping c_E from E into E'' is a topological vector space isomorphism from E onto the strong dual of E'_b. 07SJ
Proposition 4 — The strong dual of a reflexive space is reflexive. 07SK
Theorem 2 — In order that a locally convex Hausdorff space E be reflexive, it is necessary and sufficient that it is barrelled and that every bounded subset of E is relatively compact for th… 07SL
Remark 1 Let E be an infinite dimensional real hilbertian space. Let F denote the space E endowed with the weakened topology. The spaces E and F have the same dual E', and E is a reflexive… 07SM
Remark 2 Let E be a reflexive space and M a closed vector subspace of E. It may happen that neither M nor E/M are reflexive spaces (IV, p. 63, exerc. 10). \ For the case of normed spaces, s… 07SN
Proposition 5 — Let E be a normed space, E' its dual and E'' its bidual. The (closed) unit ball in E'' is the closure of the unit ball B in E for the weak topology σ(E'', E'). 07SO
Remark — A Banach space E is closed in its bidual E'' for the strong topology, but is dense for the weak topology (prop. 5). 07SP
Proposition 6 — In order that a normed space E be reflexive, it is necessary and sufficient that the unit ball in E be compact for the weakened topology σ(E, E'). 07SQ
Proposition 7 — Let E be a reflexive Banach space and M a closed vector subspace of E. Then M and E/M are reflexive Banach spaces. 07SR
Example — 1) Let ℓ^∞(N) denote the Banach space of bounded sequences x = (xₙ)_n ∈ N of scalars, with the norm 07SS
Definition 4 A locally convex Hausdorff and barrelled space in which every bounded subset is relatively compact is called a Montel space. 07ST
Example 1 Every finite dimensional Hausdorff space is a Montel space. A normed space which is a Montel space is locally compact, hence is finite dimensional (I, p. 15, th. 3). 07SU
Example 2 With the notations and hypothesis of prop. 7 of III, p. 6, the space E, being the inductive limit of Banach spaces, is barrelled (III, p. 25); moreover, every bounded subset of E i… 07SV
Example 3 Every strict inductive limit E of a sequence (Eₙ) of Montel spaces (II, p. 33) such that Eₙ is closed in Eₙ₊₁ for all n, is a Montel space; in fact, E is Hausdorff (II, p. 32, prop… ↩ 1 07SW
Example 4 \ Let U be an open set in Rⁿ and let C^∞(U) be the Fréchet space of infinitely differentiable functions on U (III, p. 9). We shall prove that this is a Montel space. Since C^∞(U) i… 07SX
Proposition 8 Let E be a Montel space and 𝔗 a filter on E, which converges to a point x₀ in E for the weakened topology. If 𝔗 is a countable base, or contains a bounded set, then 𝔗, converges to… 07SY
Proposition 9 The strong dual of a Montel space is a Montel space. 07SZ
Proposition 10 Every metrizable Montel space satisfies the first axiom of countability. 07T0
Lemma 1 Suppose the metric space X does not satisfy the first axiom of countability. Then there exists a real number ε > 0 and an uncountable subset A in X such that d(x, y) ≥ ε for all di… 07T1
Lemma 2 There exists a sequence (xₘ)_m ≥ 0 consisting of distinct elements of A, which is bounded in E. 07T2
Corollary — Let E be a metrizable Montel space. Then there exists a countable dense set in the strong dual of E. 07T3

§ 3. Dual of a Fréchet space

Proposition 1 — Let E be a locally convex space. The following conditions are equivalent : 07TU
Definition 1 — A locally convex space E is said to be semi-barrelled if it satisfies the equivalent conditions of prop. 1. 07TV
Proposition 2 — Let E be a locally convex metrizable space and F its strong dual. The space F is complete, semi-barrelled and satisfies the following condition : (DB) There exists a sequence (Aₙ… 07TW
Corollary — Let G be a Hausdorff locally convex space, and let (uₙ) be a sequence of linear mappings from F into G, converging simply to a mapping u from F into G. Then u is continuous, and… 07TX
Proposition 3 — Let E be a locally convex metrizable space, E'_b its strong dual and G a Fréchet space. The space L_b(E'_b ; G) is a Fréchet space. 07TY
Corollary — The bidual of a locally convex metrizable space is a Fréchet space. 07TZ
Proposition 4 — Let E be a reflexive Fréchet space. The strong dual E'_b of E is the inductive limit of a sequence of Banach spaces. 07U0
Corollary The strong dual of a reflexive Fréchet space is bornological and barrelled. 07U1
Theorem 1 (Banach-Dieudonné) — Let E be a locally convex metrizable space. The following topologies coincide on the dual E' of E : 07U2
Corollary 1 Let E be a locally convex metrizable space. Every precompact subset of E is contained in the closed convex balanced envelope of the set of points of a sequence converging to 0. 07U3
Corollary 2 Let E be a Fréchet space. In order that a convex subset A of the dual E' of E be closed for σ(E', E), it is necessary and sufficient that A ∩ U^∘ is closed for σ(E', E) for every n… 07U4
Corollary 3 Let E be a Banach space and M a vector subspace of the dual E' of E. In order that M be closed for the weak topology σ(E', E), it is necessary and sufficient that its intersection… 07U5
Example — Let H be a hilbertian space satisfying the first axiom of countability; let H_σ denote the space H with the weakened topology assigned to it. Let L¹(H) be the Banach space of nuc… 07U6
Lemma 1 Let E and F be two locally convex metrizable spaces, and u be a continuous linear mapping from E'_b into F. Then there exists a neighbourhood U of 0 in E'_b whose image under u is… 07U7
Theorem 2 Let E₁ and E₂ be two reflexive Fréchet spaces, and G a locally convex Hausdorff space. For i = 1, 2, let Fᵢ be the strong dual of Eᵢ. Then every separately continuous bilinear mapp… 07U8

§ 4. Strict morphisms of Fréchet spaces

Proposition 1 Let E and F be two locally convex spaces and u a continuous linear mapping from E into F. In order that u be a strict morphism, it is necessary and sufficient that the following co… 07UR
Proposition 2 Let E and F be two Hausdorff locally convex spaces and u a continuous linear mapping from E into F. In order that u be a strict morphism, it is necessary and sufficient that its tr… 07US
Remark — With the notations of prop. 2, property b') also implies that u is a strict morphism for the weakened topologies (II, p. 49, cor. 3). 07UT
Proposition 3 Let E and F be two locally convex spaces, and u a continuous linear mapping from E into F. We assume that E is Hausdorff and that F is metrizable. For u to be a strict morphism, it… 07UU
Theorem 1 Let E and F be two Fréchet spaces and u a continuous linear mapping from E into F. The following conditions are equivalent : 07UV
Lemma 1 Let E and F be two Hausdorff locally convex and quasi-complete spaces and u a continuous linear mapping from E into F. For ᵗ u to be a strict morphism from F'_c into E'_c, it is ne… 07UW
Corollary 1 Under the hypothesis of th. 1, the following conditions are equivalent : (i) u is a strict injective morphism ; (ii) ᵗ u is a strict surjective morphism for the weak topologies. (i… 07UX
Corollary 2 Under the hypothesis of th. 1, the following conditions are equivalent : (i) u is surjective ; (ii) u is a strict surjective morphism ; (iii) ᵗ u is a strict injective morphism for… 07UY
Corollary 3 — Under the hypotheses of th. 1, the following property implies the properties a) to g) : h) ᵗ u is a strict morphism from F'_b into E'_b. When E and F are both Banach spaces, or b… 07UZ
Corollary 4 — Suppose E and F are Banach spaces. For ᵗ u to be surjective, it is necessary and sufficient that there exist a real number r > 0 such that | x | ≤ r. | u(x) | for all x ∈ E. This… 07V0
Corollary 5 — Let E and F be two Fréchet spaces and u a continuous linear mapping from E into F. The following conditions are equivalent : a) u is an isomorphism from E onto F. b) u is an isom… 07V1
Proposition 4 — Let E and F be two Fréchet spaces, and u a continuous linear mapping from E into F. The following conditions are equivalent : 07V2
Corollary — Suppose E and F are Banach spaces. The following conditions are equivalent : (i) u is surjective. (ii) There exists a real number r 0, such that |f| ≤ r.|ᵗu(f)| for all f ∈ F'. (… 07V3

§ 5. Compactness criteria

Proposition 1 — Let E be a Hausdorff locally convex space, and A a subset of E. Let E_σ denote the space E with the weakened topology. a) If every infinite sequence of points of A has a limit po… 07VR
Proposition 2 Let D be a dense subset of X and A a subset of the space Cₛ(X). The following conditions are equivalent : (i) A is relatively compact in Cₛ(X). (ii) From every infinite sequence of… 07VS
Theorem 1 (Eberlein) — Let E be a Hausdorff and quasi-complete locally convex space, T a topology on E which is compatible with the duality between E and E' and A a subset of E. For A to be relatively… 07VT
Theorem 2 (Šmulian) — Let E be a Fréchet space and A a subset of E. Let E_σ denote the space E endowed with the weakened topology. The following conditions are equivalent : 07VU
Lemma 1 — Let F be a Fréchet space satisfying the first axiom of countability. Every subset C of F which is compact for the topology T induced by σ(F, F') is metrizable for this topology. 07VV
Lemma 2 Let X be a compact space. If a sequence (fₙ)_n ∈ N is bounded in C(X) and converges simply on X to a continuous function f, then μ(f) = lim_n → ∞ μ(fₙ) for every μ in C(X)'. 07VW
Proposition 3 Let X be a compact space, and let A be a bounded subset of C(X). For A to be relatively compact for the topology of simple convergence, it is necessary and sufficient that it is re… 07VX
Corollary Let S be a topological space and A a bounded subset of Cᵇ(S). The following conditions are equivalent : (i) A is relatively compact for σ(Cᵇ(S), Cᵇ(S')); (ii) if (fₙ)_n ∈ N is a se… 07VY
Theorem 3 (Krein) Let E be a Hausdorff and quasi-complete locally convex space, and let T be a topology on E compatible with the duality between E and E'. Let A be a subset of E which is compact for… 07VZ

App. 0. Fixed points of groups of affine transformations

Theorem 1 (Markoff-Kakutani) — Let E be a Hausdorff locally convex vector space over the field R, and K a non-empty compact convex subset of E. Let Γ be a set of continuous affine transformations on K, pairwis… 07WI
Corollary — Let G be a solvable group of continuous affine transformations on K. Then there exists a point in K which is invariant under G. 07WJ
Definition 1 A positive linear form μ on the space B(X; R), where X is a topological space, for which |μ| = 1, is called a mean on X. 07WK
Lemma 1 The set K of means on X is the subset of the unit ball of the dual of the Banach space E = B(X; R) whose elements are the linear forms μ such that μ(1) = 1. It is a subset of E' wh… 07WL
Proposition 1 Let X be a topological space and G a solvable group. We assume that G operates on X on the left, in such a way that for all g ∈ G, the mapping x ↦ g.x from X into X is continuous… 07WM
Corollary Let G be a solvable topological group. Then there exists a mean on G which is invariant under the left and the right translations. 07WN
Lemma 2 Suppose K contains at least two points. For every ε > 0, there exists a partition of K into two non-empty subsets K₁ and K₂, having the following properties : 07WO
Lemma 3 Let G be a group of continuous (for T) affine transformations on K. Suppose that K is non-empty and that |gx - gy| = |x - y| for all x, y in K and all g in G. Then there exists a p… 07WP
Theorem 2 (Ryll-Nardzewski) — Let E be a normed space and K a non-empty convex subset of E, which is compact for the weakened topology σ(E, E'). Let G be a group of isometric affine transformations of K. Then… 07WQ
Corollary — Let E be a reflexive Banach space, G a group of automorphisms of the normed space E, and K a subset of E. Suppose that K is non-empty, convex, closed, bounded and stable under G… 07WR

V. Hilbertian spaces (elementary theory)

§ 1. Prehilbertian spaces and hilbertian spaces

Definition 1 Let E be a vector space over the field K. A hermitian form (on the left) on E is a map f from E × E into K satisfying the following conditions (for x₁, x₂, x, y₁, y₂, y in E and λ… 07X0
Remark — We observe that formula (6) is valid for every sesquilinear form on E × E (that is to say, for every function f satisfying (1) and (2), but not necessarily (3)). This remark show… 07X1
Proposition 1 If f is a hermitian form on E, and M a vector subspace of E such that f(x, x) = 0 for all x ∈ M, then we also have f(x, y) = 0 for every pair of points x, y in M. 07X2
Definition 2 Let E be a vector space over the field K. A hermitian form f on E is said to be positive if f(x, x) ≥ 0 for all x ∈ E. 07X3
Proposition 2 — If f is a positive hermitian form, we have 07X4
Corollary 1 — If f is a positive hermitian form, the set N of all x \in E such that f(x, x) = 0 coincides with the vector subspace of all x \in E such that f(x, y) = 0 for all y \in E. 07X5
Corollary 2 — For a positive hermitian form to be separating, it is necessary and sufficient that the relation x \neq 0 implies f(x, x) 0. 07X6
Proposition 3 — Let f be a positive hermitian form on E. Put 07X7
Remark — 1) Suppose f is positive and separating, and let x, y be two vectors ≠ 0. The proof of Cauchy-Schwarz inequality shows that, if the two members of (7) are equal, then there exist… 07X8
Definition 3 Let E be a vector space over the field K. A semi-norm p on E is said to be prehilbertian if there exists a positive hermitian form f on E such that p(x) = f(x, x)^1/2 for all x ∈ E… 07X9
Definition 4 A prehilbertian space is a set E with the structure of a vector space over K and with a positive hermitian form. We say that E is a real (resp. complex) prehilbertian space when K… 07XA
Example 1 The form (λ, μ) ↦ overlineλ μ defines a prehilbertian structure on K, said to be canonical. When K is considered as a prehilbertian space, we shall always mean, unless otherwise me… 07XB
Example 2 Let I be an interval (bounded or not) in R, and let E be the set of regulated functions (FVR, II, p. 4) defined on I with values in C, having compact support. It is clear that E is… 07XC
Example 3 Let n geqslant 0 be an integer. We define a prehilbertian space structure on the space Kⁿ, by means of the hermitian form 07XD
Example 4 Let ℓ² (or ℓ²(N)) be the set of sequences x = (xₙ)_n ∈ N of elements of K such that ∑ₙ₌₀^∞ |xₙ|² is finite. One can show that ℓ² is a vector subspace of Kᴺ and define a prehilberti… 07XE
Example 5 Let E be a real prehilbertian space, f the corresponding symmetric bilinear form on E. Let E_(c) be the vector space complexification of E; we identify E with a subset of E_(c) by… 07XF
Definition 5 A hilbertian space (or Hilbert space) is a prehilbertian space which is Hausdorff and complete. We say that a norm on a vector space E (over K) is hilbertian if it is prehilbertian… 07XG
Example 1 The prehilbertian spaces defined in examples 1, 3, 4 of V, p. 4, are hilbertian spaces. On the other hand, the prehilbertian space defined in example 2 is neither Hausdorff, nor co… 07XH
Example 2 \ Let X be a Hausdorff topological space and let μ be a positive measure on X. Let L²(X, μ) be the space consisting of equivalence classes, for μ, of all square μ-integrable functi… 07XI
Example 3 \ Let n geqslant 1 be an integer and let U be an open set in Rⁿ. Let μ be the measure on U induced by the Lebesgue measure on Rⁿ, and put H⁰ = L²(U, μ). Let H¹ denote the space of… 07XJ
Example 4 \ Let X be a differential variety of class Cʳ (with r geqslant 1) pure of finite dimension n. 07XK
Example 5 \ Let D be the open disc with centre 0 and radius 1 in C. The Hardy space H²(D) consists of all holomorphic functions f : D → C for which sup_0 < R < 1 ∫₀¹ |f(R · e(θ))|² dθ < + ∞… 07XL
Proposition 4 — Let E be a Hausdorff prehilbertian space, hatE the normed space completion of E(GT, IX, § 3, No. 3). The scalar product (x, y) ↦ ⟨ x|y ⟩ extends by continuity to a positive and s… 07XM
Corollary — Let V be a vector space over K and f a positive hermitian form on V. a) There exists a Hilbert space E and a linear mapping u : V → E such that f(x, y) = ⟨ u(x)|u(y) ⟩ for x, y i… 07XN
Example 7 Let G be a group (with unit element 1) and π a homomorphism from G into the group of automorphisms of a complex hilbertian space E ; we say that π is a unitary representation of G… 07XO
Proposition 5 — Let E be a prehilbertian space. Let d be a real number > 0, δ a real number such that 0 leqslant δ leqslant d. Let B and B' be subsets of E defined by | x | < d, 07XP
Theorem 1 Let E be a prehilbertian space, and H a non-empty convex subset of E such that H is a Hausdorff and complete uniform subspace of E. For every x ∈ E, there exists a unique point p_H… 07XQ
Corollary 1 Let I be a set directed by a filter 𝔉 and let (yᵢ)_i ∈ I be a family of points of H. Let x ∈ E. Suppose that we have lim_i, 𝔉 | x - yᵢ | = inf_z ∈ H | x - z | . Then yᵢ tends to p_… 07XR
Corollary 2 For every x, y in E, we have | p_H(x) - p_H(y) | ≤ | x - y | . In particular, the mapping p_H from E into H is continuous. 07XS
Proposition 6 Let E be a prehilbertian space and let Φ be a non-empty, directed decreasing set of non-empty Hausdorff and complete convex subsets of E. For every x ∈ E and every subset H of E, p… 07XT
Proposition 7 Let E be a Hausdorff prehilbertian space and let Ψ be a non-empty directed increasing set of non-empty, convex, complete subsets of E. Put A = ⋃_H ∈ Ψ H and suppose that the closur… 07XU
Theorem 2 Let E be a prehilbertian space and M a vector subspace of E, which is Hausdorff and complete. Then E is the topological direct sum of M and of M^∘ the subspace orthogonal to M. The… 07XV
Corollary Let E be a Hausdorff prehilbertian space and M a finite dimensional vector subspace of E. Then E is the direct sum of M and M^∘. 07XW
Proposition 8 — Let E be a hilbertian space, M a closed vector subspace of E, I a non-empty ordered directed set and (Mᵢ)_i ∈ I a family of closed vector subspaces of E. We assume that either th… 07XX
Proposition 9 — Let E be a hilbertian space and M, N two closed vector subspaces of E. 07XY
Remark — Let E be a hilbertian space and M, N two closed vector subspaces of E. The relation M ⊂ N is equivalent to the orthogonality of M and N^∘, that is to say, to the relation p_M p_N… 07XZ
Theorem 3 Let E be a hilbertian space. For every x ∈ E, let x^* be the continuous linear form y ↦ ⟨ x | y ⟩ on E; the mapping x ↦ x^* is a bijective, semi-linear (for the automorphism ξ ↦ ba… 07Y0
Corollary 1 In order that the family (xᵢ)_i ∈ I of points of a hilbertian space E be total, it is necessary and sufficient that the relations ⟨ xᵢ|y ⟩ = 0 for y ∈ E and for all indices i ∈ I i… 07Y1
Corollary 2 Let E and F be two hilbertian spaces. For u ∈ L(E; F), x ∈ E and y ∈ F, put 07Y2
Theorem 4 In a hilbertian space E, the unit ball is weakly compact. 07Y3
Proposition 10 If, in a hilbertian space E, a filter 𝔉 converges weakly to x₀, and if moreover lim_𝔉 |x| = |x₀|, then 𝔉 converges to x₀ for the initial topology of E. 07Y4
Remark — If E is a Hausdorff prehilbertian space and hatE the hilbertian space completion of E, we know (III, p. 16) that the dual E' of E can be identified with the dual of hatE; it then… 07Y5

§ 2. Orthogonal families in a hilbertian space

Proposition 1 Let (Eᵢ)_i ∈ I be a family of hilbertian spaces, P the product vector space ∏_i ∈ I Eᵢ and E the subset of P consisting of all families x = (xᵢ)_i ∈ I such that ∑_i ∈ I |xᵢ|² is fi… 07Z7
Definition 1 Let (Eᵢ)_i ∈ I be a family of hilbertian spaces. The hilbertian space E defined in prop. 1 is called the external hilbertian sum of the family (Eᵢ)_i ∈ I and written as ⨁_i ∈ I Eᵢ… 07Z8
Example — Let E be a hilbertian space and I a set of indices. Let ℓ²_E(I) denote the external hilbertian sum of the family (Eᵢ)_i ∈ I where Eᵢ = E for all i ∈ I. In other words, ℓ²_E(I) is… 07Z9
Definition 2 A hilbertian space E is said to be a hilbertian sum of a family (Eᵢ)_i ∈ I of closed vector subspaces of E when : 07ZA
Theorem 1 — Let E be a hilbertian space which is a hilbertian sum of a family (Eᵢ)_i ∈ I of closed vector subspaces of E. There exists an isomorphism f and only one, from E onto the external… 07ZB
Corollary 1 — Let E be a hilbertian space, which is a hilbertian sum of a family (Eᵢ)_i ∈ I of closed vector subspaces of E; for all i ∈ I, let p_Eᵢ be the orthoprojector (V, p. 13) from E ont… 07ZC
Corollary 2 — Let E be a Hausdorff prehilbertian space, (Eᵢ)_i ∈ I a family of complete vector subspaces of E such that, for every pair of distinct indices i, k in I, the subspaces Eᵢ and Eₖ a… 07ZD
Remark Let E be a Hausdorff prehilbertian space, (Vᵢ)_i ∈ I a family of vector subspaces of E such that for every pair of distinct indices i, k, the subspaces Vᵢ and Vₖ are orthogonal. Th… 07ZE
Proposition 2 — Let E be a hilbertian space and (V_λ)_λ ∈ L a family of closed vector subspaces of E; for every λ ∈ L, let (W_λ μ)_μ ∈ M_λ be a family of closed vector subspaces of V_λ such that… 07ZF
Definition 3 In a prehilbertian space, a family (ei){i∈I} of vectors is said to be orthogonal if ei and ek are orthogonals for all i ≠ k, and is said to be orthonormal, if in addition |eᵢ| = 1… 07ZG
Proposition 3 In a Hausdorff prehilbertian space E, every orthonormal family is topologically independent. 07ZH
Proposition 4 Let E be a Hausdorff prehilbertian space, (ei){i∈I} an orthonormal family in E, V the closed vector subspace of E generated by the ei. 07ZI
Proposition 5 — Let (eᵢ)_i ∈ I be an orthonormal family in a Hausdorff prehilbertian space E. The following properties are equivalent : a) the family (eᵢ) is total ; b) for every x ∈ E, the fami… 07ZJ
Definition 4 — An orthonormal and total family in a Hausdorff prehilbertian space E is called an orthonormal basis of E. 07ZK
Example Let E be the space of all complex valued continuous functions on R, such that f(x + n) = f(x) for x ∈ R and n ∈ Z. We assign to E the scalar product defined by 07ZL
Theorem 2 For every orthonormal set L in a hilbertian space E, there exists an orthonormal basis B of E containing L. 07ZM
Corollary 1 In every hilbertian space, there exists an orthonormal basis. 07ZN
Corollary 2 Every hilbertian space is isomorphic to a space ℓ²(I). 07ZO
Proposition 6 Let E be a Hausdorff prehilbertian space, and let (aₙ)_n ∈ I (I an interval of N with origin 1) be a countable (finite or not) independent family of vectors of E. There exists an o… 07ZP
Corollary In every Hausdorff prehilbertian space E satisfying the first axiom of countability, there exists a countable orthonormal basis. 07ZQ
Example Let I be the interval (-1, 1) of R and E the vector space of real valued continuous functions on I. Let x denote the canonical injection from I into R, considered as an element of… 07ZR
Proposition 7 In a hilbertian space E, two orthonormal bases are equipotent. 07ZS
Corollary 1 Given two orthonormal bases in a hilbertian space E, there exists an automorphism of E transforming the first basis into the second. 07ZT
Corollary 2 — In order that the hilbertian spaces ℓ²(I) and ℓ²(J) be isomorphic, it is necessary and sufficient that I and J are equipotent. 07ZU

§ 3. Tensor product of hilbertian spaces

Proposition 1 — The sesquilinear form Φ on E₁ ⊗ E₂ is hermitian and positive, hence assigns the structure of a prehilbertian space to E₁ ⊗ E₂. This space is Hausdorff if E₁ and E₂ are Hausdorff. 080B
Proposition 2 For i = 1, 2, let Eᵢ and Fᵢ be two Hausdorff prehilbertian spaces and let uᵢ ∈ L(Eᵢ; Fᵢ). The linear mapping u₁ ⊗ u₂ from E₁ ⊗₂ E₂ into F₁ ⊗₂ F₂ is continuous and we have |u₁ ⊗ u₂|… 080C
Definition 1 Let E₁, ..., Eₙ be hilbertian spaces. The completion of the Hausdorff prehilbertian space E₁ hat⊗₂ ... hat⊗₂ Eₙ is called the hilbertian tensor product of the Eᵢ and is denoted by… 080D
Remark Let E₁, ..., Eₙ be Hausdorff prehilbertian spaces, and hatE₁, ..., hatEₙ their respective completions. Then E₁ ⊗₂ ... ⊗₂ Eₙ is a prehilbertian subspace of hatE₁ ⊗₂ ... ⊗₂ hatEₙ. Si… 080E
Proposition 3 Let E₁, ..., Eₙ be hilbertian spaces. Suppose that for 1 ≤ i ≤ n the space Eᵢ is a hilbertian sum of a family (E_i,α)_α ∈ A(i) of closed vector subspaces. Then E₁ ⊗₂ ... ⊗₂ Eₙ is a… 080F
Corollary 1 — For 1 ≤ i ≤ n, let (e_i,α)_α ∈ A(i) be an orthonormal basis of Eᵢ. Then the family of vectors e_1,α₁ ⊗ … ⊗ e_n,αₙ as (α₁, …, αₙ) ranges over A(1) × ⋯ × A(n) is an orthonormal bas… 080G
Corollary 2 — Let E₁ and E₂ be two hilbertian spaces, and (eᵢ)_i ∈ I an orthonormal basis of E₁. Let (yᵢ)_i ∈ I be a family of elements of E₂, such that ∑_i ∈ I | yᵢ | ² < + ∞. Then the family… 080H
Example 1 By cor. 1, the space ℓ²(I) hat⊗₂ ℓ²(J) is canonically isomorphic to ℓ²(I × J), the tensor product x ⊗ y of x = (xᵢ)_i ∈ I and y = (yⱼ)_j ∈ J can be identified with the family (xᵢ y… 080I
Example 2 \ Let X be a Hausdorff topological space, and μ a positive measure on X. Let E be a hilbertian space. We can identify L²(X, μ) hat⊗₂ E with L²_E(X, μ) in a canonical way : if f is… 080J
Proposition 4 Let (eᵢ)_i ∈ I be an orthonormal basis of the hilbertian space E. For every α in N⁽ᴵ⁾, put 080K
Corollary — Suppose that the hilbertian space E is the direct sum of the orthogonal subspaces M and N. The canonical isomorphism g from S(M) ⊗ S(N) onto S(E) (A, III, § 6, No. 6) extends uni… 080L
Example 1 Let d ≥ 1 be an integer and ω a positive function on Rᵈ, locally integrable with respect to the Lebesgue measure μ. Let E be the hilbertian space L²(Rᵈ, ω.μ), and let S = S(E). The… 080M
Example 2 Let X be a Hausdorff topological space, μ a positive measure of norm 1 on X and E a hilbertian subspace of the real hilbertian space L²_R(X, μ). We say that E is a gaussian space i… 080N
Example — With the notations of example 1 of V, p. 32, we can identify the hilbertian space hatΛ(E) with the set of all sequences (fₙ)_n ≥ 0 of measurable functions for which the number |f… 080O
Proposition 5 Let (eᵢ)_i ∈ I be an orthonormal basis of the hilbertian space E. Endow I with a totally ordered structure. Then the set of all elements e_i₁ ∧ ... ∧ e_iₙ for i₁ < ... < iₙ is an o… 080P
Corollary Suppose that the hilbertian space E is the direct sum of two orthogonal subspaces M and N. The canonical isomorphism g from Λ(M) ⊗ Λ(N) onto Λ(E) (A, III, § 7, No. 7) extends in a… 080Q
Proposition 6 Let E be a hilbertian space. We have 080R
Corollary 1 a) Let x₁, …, xₙ be elements of the hilbertian space E. We have 080S
Corollary 2 — Let (aᵢⱼ)_1 ≤ i,j ≤ n be a hermitian matrix, with complex elements, and with determinant D. Suppose that the inequality 080T
Corollary 3 (« Hadamard’s inequalities ») — Let (aᵢⱼ)_1 ≤ i, j ≤ n be a matrix with complex elements, and with determinant D. Put 080U

§ 4. Some classes of operators in hilbertian spaces

Proposition 1 — Let E and F be two hilbertian spaces. For every mapping u ∈ L(E; F), there exists a unique mapping u^* ∈ L(F; E) such that 080X
Definition 1 — Let E and F be two hilbertian spaces. For every continuous linear mapping u : E → F, the continuous linear mapping from F into E defined by formula (1) is called the adjoint of u… 080Y
Proposition 2 For every u ∈ L(E ; F), we have 080Z
Remark 1 Suppose the hilbertian space E does not reduce to 0. We identify L(K;E) with E by the mapping u ↦ u(1); in other words, the vector x of E is identified with the mapping λ ↦ λ.x fro… 0810
Proposition 3 — Let E and F be two hilbertian spaces and u ∈ L(E;F). The following conditions are equivalent : 0811
Remark 2 — The relation u^*u = 1_E does not characterize all the automorphisms of the hilbertian space E. For example, let E = ℓ²(N) and let u be defined by u(xₙ) = xₙ₋₁ for n ≥ 1 and u(x)₀… 0812
Remark 3 — The definition (1) of the adjoint u^* can also be written as ⟨ y|u(x)⟩ = ⟨ u^*(y)|x⟩ , or, by V, p. 15, as ⟨ u(x), y^*⟩ = ⟨ x, (u^*(y))^*⟩ . But we also have ⟨ u(x), y^*⟩ = ⟨ x… 0813
Definition 2 Let E and F be two hilbertian spaces and u ∈ L(E; F). The orthogonal of the kernel of u in E is said to be the initial subspace of u and the closure of the image of u in F is calle… 0814
Proposition 4 (i) The initial (resp. final) subspace of u^* is equal to the final (resp. initial) subspace of u. 0815
Definition 3 Let E and F be two hilbertian spaces. A mapping u ∈ L(E; F) is said to be partially isometric if |u(x)| = |x| for all x belonging to the initial subspace of u. 0816
Proposition 5 Let u ∈ L(E; F), let P be its initial subspace and Q be the final subspace. Let p (resp. q) denote the initial (resp. final) orthoprojector of u. Assume that u is partially isometr… 0817
Proposition 6 For every u ∈ L(E; F), the following conditions are equivalent : 0818
Definition 4 Let E be a hilbertian space and u ∈ L(E). We say that u is normal if it commutes with its adjoint u^*. 0819
Proposition 7 For u ∈ L(E) to be normal, it is necessary and sufficient that |u(x)| = |u^*(x)| for all x ∈ E. ↩ 1 081A
Proposition 8 Suppose that u ∈ L(E) is normal. Let N be the kernel of u and M the orthogonal of N in E; let m and n be two positive integers such that m + n ≥ 1. Then N is the kernel of uᵐ(u^*)ⁿ… 081B
Corollary Let λ ∈ K. The following subspaces of E are equal : a) the eigen subspace of u relative to λ; b) the eigen subspace of u^* relative to overlineλ; 081C
Definition 5 Let E be a hilbertian space and let u ∈ L(E). We say that u is hermitian if u^* = u. 081D
Proposition 9 For every hermitian endomorphism u of E, we have 081E
Proposition 10 — Suppose K is C. Let u ∈ L(E). Then there exists a unique pair (h₁, h₂) of hermitian endomorphisms of E, such that u = h₁ + ih₂. In order that u is normal, it is necessary and suf… 081F
Proposition 11 — Let p ∈ L(E). In order that p is the orthoprojector from E onto a closed vector subspace of E, it is necessary and sufficient that p² = p = p^*. 081G
Definition 6 — Let E be a hilbertian space and u ∈ L(E). We say that u is positive, and write u ≥ 0, if u is hermitian and if ⟨ x|u(x) ⟩ ≥ 0 for all x ∈ E. 081H
Proposition 12 — Let u be a hermitian (resp. positive) element of L(E) and let v be a continuous linear mapping from E into a hilbertian space F. Then vuv^* is a hermitian (resp. positive) elemen… 081I
Remark 1 For every hermitian u in L(E), put m(u) = inf_|x|=1 ⟨ x|u(x)⟩, M(u) = sup_|x|=1 ⟨ x|u(x)⟩. If E is not just 0, m(u) and M(u) are finite; moreover, M(u) is the smallest real number… ↩ 1 081J
Remark 2 Let M and N be two closed vector subspaces of E, and p_M (resp. p_N) the orthoprojector from E onto M (resp. N). Then M ⊂ N if and only if p_M ≤ p_N. For, we have p_M^* p_M = p_M… 081K
Proposition 13 — Let H(E) be the set of all continuous hermitian endomorphisms of the hilbertian space E. Let F be a non-empty, directed increasing and bounded subset of H(E). 081L
Lemma 1 There exists an isomorphism θ from the vector space F ⊗ E' onto the space L_f(E; F) of all finite rank continuous linear mappings from E into F, characterized by θ(b ⊗ a^*) = ba^*… 081M
Lemma 2 Let (eᵢ)_i ∈ I be an orthonormal basis of E. Then 081N
Lemma 3 Let u be a continuous and positive endomorphism of E, and F the set of all finite rank orthoprojectors on E. Then for every orthonormal basis (eᵢ)_i ∈ I of E, we have (in R₊) the e… 081O
Definition 7 Let u be a continuous and positive endomorphism of the hilbertian space E. Let Tr(u) = sup_p ∈ F τ(p u p) (upper bound in overlineR₊), where F is the set of all finite rank orthopr… 081P
Definition 8 — Let E be a complex hilbertian space. We write L¹(E) for the vector subspace of L(E) generated by all continuous, positive endomorphisms of E with finite trace. 081Q
Proposition 14 — Let E₁, ..., Eₙ be hilbertian spaces, E = E₁ hat⊗₂ ... hat⊗₂ Eₙ, and uᵢ a continuous endomorphism of Eᵢ for 1 ≤ i ≤ n. If u₁, ..., uₙ are positive, then so is u = u₁ hat⊗₂ ... ha… 081R
Remark 2 — Let E be a hilbertian space, which is the hilbertian sum of a family (Eᵢ)_i ∈ I of hilbertian subspaces. Let u be an element of L(E) such that u(Eᵢ) ⊂ Eᵢ for all i ∈ I; let uᵢ be… 081S
Definition 9 Let E and F be two hilbertian spaces. A continuous linear mapping u from E into F is called a Hilbert-Schmidt mapping if the trace of the positive endomorphism u^*u of E is finite… 081T
Theorem 1 Let E and F be two hilbertian spaces. (i) The set L²(E, F) is a vector subspace of L(E; F) and u ↦ |u|₂ is a hilbertian norm (V, p. 6) on L²(E; F). (ii) The isomorphism θ from F ⊗… 081U
Example 1 Let I and J be two sets. By the proof given above, in order that a mapping u from ℓ²(I) into ℓ²(J) be a Hilbert-Schmidt mapping, it is necessary and sufficient that there exists a… 081V
Example 2 \ Let X and Y be two Hausdorff topological spaces, endowed respectively with positive measures μ and ν. We can show that the Hilbert-Schmidt mappings from L²(X) into L²(Y) correspo… 081W
Remark 1 Suppose K = C. Let u and v be in L²(E; F). We have the relation 4 u^* v = ∑_ε⁴ = 1 overlineε(u + ε v)^* (u + ε v), hence u^* v belongs to L¹(E). The scalar product in the hilbertia… 081X
Remark 2 Suppose K = R. We leave it to the reader to verify that formulas (40) and (41) are again valid, and to show that L²(E; F) can be identified with the dual of L²(F; E) by means of th… 081Y
Theorem 2 Let E and F be two hilbertian spaces and u a Hilbert-Schmidt mapping from E into F. There exists an orthonormal basis (eᵢ)_i ∈ I of E which is transformed by u into an orthogonal f… 081Z
Lemma 4 The function Q : B → R is continuous. 0820
Lemma 5 Let E₁ be a closed vector subspace of E, stable under u^*u. If E₁ ≠ 0, then there exists a vector of norm 1 in E₁ ∩ P. 0821
Corollary 1 — Let v be a continuous, positive endomorphism with finite trace of the hilbertian space E. There exists an orthonormal basis (eᵢ)_i ∈ I of E and a summable family of positive real… 0822
Corollary 2 Let E be a hilbertian space. Then L¹(E) ⊂ L²(E). 0823
Corollary 3 Let v be a continuous positive endomorphism of the hilbertian space E with a finite trace. There exists a positive Hilbert-Schmidt endomorphism w of E such that v = w² and such tha… 0824
Remark — With the notations of th. 2, let J be the set of all i ∈ I such that u(eᵢ) ≠ 0. For all i ∈ J, let λᵢ = | u(eᵢ) | and fᵢ = λᵢ⁻¹ u(eᵢ). Then (eᵢ)_i ∈ J (resp. (fᵢ)_i ∈ J) is an or… 0825
Remark — 1) For every subspace F of E, let Q_F denote the restriction of Q to F and H_F that of H. We have Tr(Q_F/H_F) ≤ Tr(Q/H) and Tr(Q/H) is the upper bound of the set of all numbers T… 0826
Proposition 15 Suppose that there exists a real hilbertian space structure on E such that H(x) = |x|² for all x ∈ E. For Tr(Q/H) to be finite, it is necessary and sufficient that there exists a c… 0827
Remark 3 Let E and F be two hilbertian spaces and v a linear, not necessarily continuous mapping from E into F. Let H(x) = |x|² and Q(x) = |v(x)|² for all x ∈ E. It follows from prop. 15 th… 0828
Remark 4 Suppose E is finite dimensional. When the quadratic form H is invertible, prop. 15 applies. Let (e₁, …, eₙ) be a basis of E. Put qᵢⱼ = ⟨ eᵢ|eⱼ⟩_Q and hᵢⱼ = ⟨ eᵢ|eⱼ⟩_H and introduce… 0829