Index of statements

Most cited statements

1. Theorem 1 INT IV § 5 Let X be a locally compact space, μ a measure on X, (Fₙ) a sequence of topological spaces, F = ∏ₙ Fₙ their product. For each index n, let fₙ be a measurable mapping of X into Fₙ, a… ↩ 26 08UR
2. Theorem 1 TG I § 2 Let f be a mapping of a topological space X into a topological space X'. Then the following statements are equivalent: ↩ 25 0B05
3. Theorem 2 (Lebesgue–Nikodym) INT V § 5 — Let μ and ν be two positive measures on T. The following properties are equivalent: ↩ 20 092Y
4. Theorem 1 TG I § 10 Let f : X → Y be a continuous mapping. Then the following four statements are equivalent: ↩ 19 0BAE
5. Lemma 1 INT III § 1 Let X be a locally compact space, K a compact subset of X, and (Vₖ)_1 ≤ k ≤ n a finite covering of K by open sets of X. Then, there exist n continuous mappings fₖ of X into [0, 1]… ↩ 19 08L8
6. Theorem 1 INT III § 4 — Let X and Y be two locally compact spaces, λ a measure on X, μ a measure on Y; there exists one and only one measure ν on X × Y such that, for every function g ∈ K(X; C) and ever… ↩ 19 08O8
7. Proposition 4 A II § 2 Let P be an A-module. The following properties are equivalent: ↩ 18 05N5
8. Theorem 5 INT IV § 5 — In order that a mapping f of X into a Banach space F be p-th power integrable (1 ≤ p < +∞), it is necessary and sufficient that f be measurable and that Nₚ(f) be finite. ↩ 18 08VI
9. Theorem 2 TG I § 9 If f is a continuous mapping of a quasi-compact space X into a topological space X', then the set f(X) is quasi-compact. ↩ 17 0B8C
10. Proposition 4 A II § 3 If E is a right A-module, the mapping h : x ↦ x ⊗ 1 of E into E ⊗_A (sA_d) is a right A-module isomorphism, whose inverse isomorphism g is such that g(x ⊗ λ) = xλ for x ∈ E, λ ∈ A. ↩ 16 05P0
11. Theorem 1 A VIII § 3 Let K be an algebraically closed commutative field, A a K-algebra, and M a simple A-module. Suppose that the dimension of M as a vector space over K is finite or, more generally, s… ↩ 16 003E
12. Proposition 2 TG I § 8 Let f, g be two continuous mappings of a topological space X into a Hausdorff space Y; then the set of all x ∈ X such that f(x) = g(x) is closed in X. ↩ 16 0B68
13. Theorem 1 AC II § 2 The ring S⁻¹A is a flat A-module (Chapter I, § 2, no. 1, Definition 2). ↩ 16 0465
14. Proposition 11 AC II § 2 Let A be a ring and S a multiplicative subset of A. For every ideal b' of S^{-1}A, let b = (iA^S)^{-1}(b') be such that b' = S^{-1}b. ↩ 16 0469
15. Theorem 4 A I § 4 Let G be a group with operators and N a normal stable subgroup of G. ↩ 15 059C
16. Proposition 7 A II § 3 The canonical mapping (23) is bijective. ↩ 15 05PA
17. Theorem 7 A II § 7 Let E be a vector space over a field K, F a subspace of E and F' the orthogonal of F in E. ↩ 15 05SI
18. Proposition 5 TG I § 10 Let f : X → X' and g : X' → X'' be two continuous mappings. a) If f and g are proper, then g ∘ f is proper. b) If g ∘ f is proper and f is surjective, then g is proper. c) If g ∘ f… ↩ 15 0BA7
19. Proposition 2 INT IX § 1 a) Let w be a positive premeasure. For every compact subset K of T, the encumbrance (w^bullet)_K induced by w^bullet on K is equal to (w_K)^bullet. For every function f ∈ F₊(T), on… ↩ 15 09MU
20. Theorem 1 AC II § 5 Let A be a ring and P an A-module. The following properties are equivalent: ↩ 15 04BJ
21. Theorem 1 AC V § 2 Let h : A → A' be an injective ring homomorphism such that A is integral over A and p a prime ideal of A. There exists a prime ideal p' of A' lying above p. ↩ 15 04PR
22. Proposition 1 A V § 9 — Let E be an extension of K contained in Ω and u a K-homomorphism of E into Ω. ↩ 14 06LK
23. Theorem 1 A VIII § 6 Let A and B be k-algebras and P an (A, B)ₖ-bimodule. Denote the (B, A)ₖ-bimodule Hom_A(P, Aₛ) by P^*. The following properties are equivalent: ↩ 14 0072
24. Theorem 1 A VIII § 14 Let A be a K-algebra of finite degree. The following properties are equivalent: ↩ 14 00GJ
25. Proposition 15 INT IV § 5 Let X be a locally compact space, μ a measure on X, A a μ-measurable subset of X, and f a mapping of A into a topological space F. The following conditions are equivalent: ↩ 14 08VZ
26. Theorem 2 LIE VI § 1 (i) The group W(R) acts simply-transitively on the set of chambers. ↩ 14 0AUV
27. Theorem 2 A V § 3 — Let E be a finitely generated extension of K, generated by the elements a₁, ..., aₘ, which are algebraic over K; then E is an extension of finite degree of K. If the degree of aᵢ… ↩ 13 06HS
28. Theorem 1 (Wedderburn) A VIII § 7 A ring is simple if and only if it is isomorphic to a matrix ring Mᵣ(D), where rgeqslant 1 is an integer and D a field. ↩ 13 00S3
29. Proposition 1 TG I § 8 Let X be a topological space. Then the following statements are equivalent: ↩ 13 0B65
30. Proposition 2 INT V § 5 Let θ be a complex measure, and u a locally θ-integrable complex function; then ↩ 13 092G
31. Proposition 1 LIE VIII § 6 Let V be a 𝔤-module, v a primitive element of V and ω the weight of v. Assume that V is generated by v as a 𝔤-module. ↩ 13 012O
32. Theorem 1 A VII § 4 — Let L be a free module over a principal ideal domain A, and let M be a submodule of L of finite rank n. Then there exist a basis B of L, n elements eᵢ of B, and n nonzero element… ↩ 12 073T
33. Theorem 1 INT IV § 1 Let H be a nonempty set of functions in J₊, directed for the relation ≤. For every positive measure μ on X, μ^*( sup_g ∈ H g ) = sup_g ∈ H μ^*(g) = lim_g ∈ H μ^*(g). ↩ 12 08P7
34. Proposition 4 INT V § 2 — Let μ be a positive measure on a locally compact space T, and let 𝔎 be a μ-dense set of compact subsets of T. There exists a summable family (μ_α)_α ∈ A of positive measures on T… ↩ 12 090Q
35. Proposition 1 AC I § 3 Let E be a right A-module. The following four properties are equivalent: ↩ 12 0444
36. Proposition 2 AC II § 2 Let A, B be two rings, S a subset of A, T a subset of B and f a homomorphism from A to B such that f(S) ⊂ T. There exists a unique homomorphism f' from A[S⁻¹] to B[T⁻¹] such that f… ↩ 12 045N
37. Proposition 2 AC V § 1 Let A, A' be two rings, R an A-algebra, R an A'-algebra (not necessarily commutative) and f : A → A' and g : R → R' two ring homomorphisms such that the diagram ↩ 12 04N0
38. Proposition 1 AC V § 2 Let h : A → A' be a ring homomorphism such that A' is integral over A, 𝔭' a prime ideal of A' and 𝔭 = h⁻¹(𝔭'). For 𝔭 to be maximal, it is necessary and sufficient that 𝔭' be so. ↩ 12 04PL
39. Theorem 2 LIE IX § 2 Let G be a connected compact Lie group. ↩ 12 01A2
40. Theorem 2 ("Zorn's lemma") E III § 2 — Every inductive ordered set has a maximal element. ↩ 11 03QT
41. Proposition 2 E III § 4 Let n be an integer. Then every cardinal α such that α ≤ n is an integer. If n ≠ 0, there exists a unique integer m such that n = m + 1, and the relation α < n is equivalent to α ≤… ↩ 11 03RB
42. Lemma 1 E III § 7 Let (E_α, f_βα) be a direct system of sets, E = varinjlim E_α its direct limit, and for each α ∈ I let f_α colon E_α → E the canonical mapping. ↩ 11 03OK
43. Proposition 6 A II § 1 Let (Eᵢ)_i ∈ I be a family of A-modules, M an A-module and, for all i ∈ I, let fᵢ : Eᵢ → M be a linear mapping. Then there exists one and only one linear mapping g : ⨁_i ∈ I Eᵢ → M… ↩ 11 05K7
44. Corollary 3 A II § 1 Let E be a free A-module, (aₜ)_t ∈ T a basis of E, F an A-module and (bₜ)_t ∈ T a family of elements of F. There exists one and only one linear mapping f : E → F such that f(aₜ) =… ↩ 11 05LG
45. Proposition 25 A II § 1 Let A, B be two rings, ρ : B → A a ring homomorphism and E an A-module. Let (α_λ)_λ ∈ L be a generating system (resp. free family of elements, basis) of A considered as a left B-mo… ↩ 11 05LX
46. Theorem 2 A II § 7 Given a generating system S of a vector space E over a field K and a free subset L of E contained in S, there exists a basis B of E such that L ⊂ B ⊂ S. ↩ 11 05RJ
47. Proposition 1 A VIII § 10 Let A be a left Artinian ring. The radical 𝔯 of A is the largest nilpotent two-sided ideal of A, and the ring A/𝔯 is semisimple. ↩ 11 00C5
48. Proposition 4 TG I § 2 Let X be a set, let (Yᵢ)_i ∈ I be a family of topological spaces, and for each i ∈ I let fᵢ be a mapping of X into Yᵢ. Let S be the set of subsets of X of the form overlinefᵢ⁻¹(Uᵢ)… ↩ 11 0B0I
49. Proposition 1 TG I § 4 Let f = (fᵢ) be a mapping of a topological space Y into a product space X = ∏_i ∈ I Xᵢ. Then f is continuous at a point a ∈ Y if and only if fᵢ is continuous at a for each i ∈ I. ↩ 11 0B29
50. Proposition 4 TG I § 11 Let A be a connected subset of a topological space X, and let f be a continuous mapping of X into a topological space X'. Then f(A) is connected. ↩ 11 0BBQ
51. Theorem 2 INT V § 4 Let (π, g) be a μ-adapted pair, and let ν = ∫ g(t) ε_π(t) dμ(t). Let f be a function defined on X, with values in a Banach space F or in overlineR. For f to be essentially ν-integr… ↩ 11 0929
52. Lemma 1 AC I § 2 Let M' xrightarrowv M xrightarroww M'' → 0 be an exact sequence of left A-modules and E a right A-module. Then the sequence ↩ 11 042F
53. Theorem 1 AC II § 3 Let M, N be two A-modules, u : M → N an A-homomorphism and, for all m ∈ Ω, let uₘ : M, → Nₘ be the corresponding Aₘ-homomorphism (§ 2, no. 2, Remark 5). For u to be injective (resp… ↩ 11 048A
54. Proposition 21 A II § 1 Every exact sequence of A-modules ↩ 10 05LM
55. Corollary 1 A V § 3 — Let E be an extension of K and A a subset of E consisting of elements that are algebraic over K. Then K(A) is algebraic over K and we have K[A] = K(A). ↩ 10 06HT
56. Theorem 1 (Jacobson) A VIII § 9 The radical of a ring A consists of the elements x∈ A such that 1 +ax is left invertible (I, §2, No. 3, p. 15) for every a∈ A. It is also the largest two-sided ideal 𝔞 such that 1… ↩ 10 00S5
57. Proposition 3 A VIII § 14 Let A be a central simple K-algebra of finite degree, and let L be a semisimple commutative subalgebra of A. The following properties are equivalent: ↩ 10 00H2
58. Corollary 1 (Principle of extension of identities) TG I § 8 Let f, g be two continuous mappings of a topological space X into a Hausdorff space Y. If f(x) = g(x) at all points of a dense subset of X, then f = g. ↩ 10 0B69
59. Lemma 2 TG III § 2 If a topological group G operates continuously on a topological space X, then the equivalence relation R defined by G is open. ↩ 10 0BI4
60. Theorem 4 INT IV § 4 — For a set A to be integrable, it is necessary and sufficient that, for every ε > 0, there exist an integrable open set G and a compact set K, such that K ⊂ A ⊂ G and |μ|(G - K) =… ↩ 10 08SY
61. Proposition 7 INT IV § 5 Let f be a measurable mapping of X into a topological space F; the inverse image under f of every closed (resp. open) set in F is measurable. ↩ 10 08V6
62. Theorem 1 INT V § 3 — Let f be a function with values in a Banach space F or in overlineR, and let H be the set of t ∈ T for which f is not λₜ-integrable. ↩ 10 0919
63. Proposition 3 INT V § 5 For every numerical function f geqslant 0 defined on T, ↩ 10 092J
64. Corollary 2 INT V § 5 Let u₁ and u₂ be two locally θ-integrable complex functions. In order that u₁ · θ = u₂ · θ, it is necessary and sufficient that u₁ and u₂ be equal locally almost everywhere. ↩ 10 092L
65. Theorem 1 INT V § 5 Let f be a function defined on T, with values in a Banach space F or in overlineR. In order that f be essentially integrable for the measure η = u · θ, it is necessary and sufficie… ↩ 10 092P
66. Proposition 2 AC I § 1 Suppose that in the commutative diagram (10) the sequences (u, v) and (u', u') are exact. Then: ↩ 10 0424
67. Proposition 4 AC V § 1 Let A be a ring, R an A-algebra (not necessarily commutative) and (xᵢ)_1 ≤ i ≤ n a finite family of elements of R which are pairwise permutable. If, for all i, xᵢ is integral over… ↩ 10 04N5
68. Theorem 1 LIE I § 5 Let g be a Lie algebra, r its radical and s its nilpotent radical. Then s = Dg ∩ r. ↩ 10 09YA
69. Theorem 4 LIE I § 6 Let g be a Lie algebra, r its radical, ρ a finite-dimensional representation of g, g' = ρ(g) and r' = ρ(r). Then the following conditions are equivalent: (a) ρ is semi-simple; (b)… ↩ 10 0A02
70. Theorem 1 LIE VII § 3 Assume that k is algebraically closed. Let 𝔤 be a Lie algebra. The group E is normal in Aut(𝔤) and operates transitively on the set of Cartan subalgebras of 𝔤. ↩ 10 00V8
71. Theorem 1 LIE VIII § 8 Let I(𝔤^*) be the algebra of invariant polynomial functions on 𝔤. Let i:S(𝔤^*)→ S(𝔥^*) be the restriction homomorphism. ↩ 10 015A
72. Theorem 3 E III § 2 Let E and F be two well-ordered sets. Then at least one of the following two statements is true : ↩ 9 03K6
73. Proposition 15 A II § 1 Given an exact sequence of A-modules (33) 0 longrightarrow E xrightarrowf F xrightarrowg G longrightarrow 0 the following propositions are equivalent: (a) The submodule f(E) of F i… ↩ 9 05KZ
74. Corollary 1 A II § 3 If the left A-module F admits a basis (b_μ)_μ ∈ M, every element of E ⊗_A F can be written uniquely in the form ∑_μ (x_μ ⊗ b_μ), where x_μ ∈ E and the family (x_μ) has finite suppo… ↩ 9 05PB
75. Corollary 2 A II § 3 If E and F are two free modules over a commutative ring C and (aλ) (resp. (bμ)) is a basis of the C-module E (resp. F), then (aλ ⊗ bμ) is a basis of the C-module E ⊗C F. ↩ 9 05PC
76. Proposition 2 A II § 5 Let ρ : A → B,\ σ : B → C be ring homomorphisms. For every left A-module E, there exists one and only one C-homomorphism ↩ 9 05QG
77. Proposition 4 A II § 5 Let E be an A-module admitting a basis (a_λ)_λ ∈ L; if φ : x ↦ 1 ⊗ x is the canonical mapping of E into ρ^*(E), then (φ(a_λ))_λ ∈ L is a basis of ρ^*(E). If ρ is injective, so is φ… ↩ 9 05QI
78. Proposition 14 A II § 7 Given an exact sequence ↩ 9 05SY
79. Theorem 2 (Steinitz) A V § 4 — Let K be a field; then there exists an algebraic closure of K. If Ω and Ω' are two algebraic closures of K, there exists a K-isomorphism of Ω onto Ω'. ↩ 9 06IS
80. Theorem 2 A VIII § 16 Let G be a finite group, and let L be a commutative K-algebra of finite degree endowed with an action of G given by an injective homomorphism λ from G to Aut_K(L). The following pr… ↩ 9 00J1
81. Corollary 1 TG I § 9 Let f be a continuous mapping of a topological space X into a Hausdorff space X'. Then the image under f of any quasi-compact (resp. relatively quasi-compact) set in X is a compact… ↩ 9 0B8D
82. Corollary 2 TG I § 9 Every continuous mapping f of a quasi-compact space X into a Hausdorff space X' is closed. If also f is bijective, then f is a homeomorphism. ↩ 9 0B8E
83. Proposition 2 TG I § 10 Let f : X → Y be a continuous injection. Then the following three statements are equivalent: a) f is proper. b) f is closed. c) f is a homeomorphism of X onto a closed subset of Y. ↩ 9 0BA4
84. Corollary 1 TG III § 4 If A is a closed (resp. compact) subset of X, then K.A is closed in X (resp. compact if X is Hausdorff). ↩ 9 0BKV
85. Proposition 3 INT III § 4 Let λ ∈ M(X; C),\ μ ∈ M(Y; C). Then ↩ 9 08OI
86. Proposition 5 INT V § 3 Let f be a ν-measurable positive numerical function defined on X, and let N be the set of t ∈ T such that f is not both λₜ-measurable and λₜ-moderated. ↩ 9 0917
87. Theorem 1 INT V § 6 — Let f be a function defined on X with values in overlineR or in a Banach space F. For f to be essentially ν-integrable, it is necessary and sufficient that f ∘ π be essentially μ… ↩ 9 094Y
88. Proposition 8 INT VI § 1 Let μ be a bounded positive measure on T, S a μ-measurable set carrying μ, f a μ-measurable mapping of T into F, such that f(S) is contained in a complete, bounded, balanced convex… ↩ 9 098S
89. Proposition 8 INT IX § 2 Let T and X be two topological spaces, π a mapping of T into X. ↩ 9 09P1
90. Proposition 1 AC I § 2 Let E be a right A-module. The following three properties are equivalent: ↩ 9 042L
91. Example 3 AC III § 2 More generally, let A be a commutative ring, b an ideal of A and A be given the b-adic filtration. If we write B=gr₀(A), F=gr₁(A)=b/b², we know (Algebra, Chapter III), that the ide… ↩ 9 04DM
92. Proposition 3 AC IV § 1 Let A be a ring, M an A-module and N a submodule of N. Then Ass(N) ⊂ Ass(M) ⊂ Ass(N) ∪ Ass(M/N). The inclusion Ass(N) ⊂ Ass(M) is obvious. Let p ∈ Ass(M), E be a submodule of M iso… ↩ 9 04JD
93. Theorem 2 AC V § 2 Let A' be a ring, G a finite group operating on A' and A the ring of invariants of G, so that A' is integral over A (§ 1, no. 9, Proposition 22). ↩ 9 04Q2
94. Proposition 1 LIE I § 6 Let g be a semi-simple Lie algebra and φ a finite-dimensional faithful representation of g. Then the bilinear form on g associated with φ is non-degenerate. ↩ 9 09ZE
95. Proposition 2 LIE VII § 1 Let V,V',W be vector spaces. Let r : S → End(V), r': S→ End(V') and q: S→ End(W) be maps. ↩ 9 00SK
96. Proposition 10 LIE VII § 1 Let 𝔤 be a Lie algebra, 𝔥 a nilpotent subalgebra of 𝔤. Denote by P the set of maps from 𝔥 to k. ↩ 9 00T0
97. Corollary LIE VIII § 1 Let E be a finite dimensional 𝔰𝔩(2, k)-module. ↩ 9 00Y0
98. Corollary 1 LIE IX § 1 Let G be a connected compact Lie group. Then C(G)₀ is a torus, D(G) is a connected compact semi-simple Lie group and the morphism (x, y) → xy from C(G)₀× D(G) to G is a finite cove… ↩ 9 019J
99. Proposition 6 E III § 7 Let I be a directed set, let (E_α, f_βα) be a direct system of sets relative to I, let E = varinjlim E_α be the direct limit, and for each α ∈ I let ↩ 8 03RR
100. Theorem 2 A I § 8 Let A be a ring and a a two-sided ideal of A. (a) The canonical mapping π of A onto A/a is a ring homomorphism. (b) Let B be a ring and f a homomorphism of A into B. If f(a) = 0, t… ↩ 8 05HI