§ 1. Semi-norms

Throughout this paragraph, $\mathbf{K}$ denotes a non-discrete valued division ring.

1. Definition of semi-norms

Definition 1

Let E be a left vector space over $\mathbf{K}$ . A mapping $p$ of E in $\mathbf{R}_+ = \{ 0, + \infty \}$ , is called a semi-norm on E if it satisfies the following axioms : (SN_I) If $x \in \mathrm{E}$ and $\lambda \in \mathbf{K}$ then $p(\lambda x) = |\lambda| \, p(x)$ . (SN_II) If $x \in \mathrm{E}$ and $y \in \mathrm{E}$ then $p(x + y) \leq p(x) + p(y)$ .

Since $p(x) \leq p(y) + p(x - y)$ and $p(y) \leq p(x) + p(y - x)$ , from $p(y - x) = p(x - y)$ , we deduce

|p(x) - p(y)| \leq p(x - y) .

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).

Example 2

For every linear form $f$ on E, the function $x \mapsto |f(x)|$ is a semi-norm on E.

Example 3

If $p_i (1 \leq i \leq n)$ is a finite set of semi-norms on E, then clearly $p'(x) = \sup_{1 \leq i \leq n} p_i(x)$ and $p''(x) = \sum_{i=1}^n \alpha_i p_i(x)$ (where the $\alpha_i$ are $\geq 0$ ) are both semi-norms on E.

A mapping $p$ of E in $\mathbf{R}_+$ is called an ultra-semi-norm if it satisfies (SN $_I$ ) and the following axiom :

(SN $_{II}'$ ) *If $x \in E$ and $y \in E$ , then $p(x + y) \leq \sup(p(x), p(y))$ .

Clearly an ultra-semi-norm is a semi-norm.

To say that the absolute value on K is ultrametric (CA, VI, § 6.2) means that it is an ultra-semi-norm on the left vector space $K_s$ , which is not identically zero.

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. c) p is uniformly continuous. d) For each real number $\alpha > 0$ , the set W(p, $\alpha$ ), of those $x \in E$ for which $p(x) < \alpha$ , is open in E. e) There exists a real number $\alpha > 0$ , such that W(p, $\alpha$ ) is a neighbourhood of 0 in E. f) *For every real number $\alpha > 0$ , the set V(p, $\alpha$ ), of those $x \in E$ for which $p(x) \leq \alpha$ , is a neighbourhood of 0 in E.

In fact, the implications c) $\Rightarrow$ a) $\Rightarrow$ b) $\Rightarrow$ d) $\Rightarrow$ e) $\Rightarrow$ f) $\Rightarrow$ c) follow immediately from (SN $_I$ ) and inequality (1).

Corollary

If p is a continuous semi-norm on E and q is a semi-norm such that $q \leq p$ , then q is continuous in E.

When p is an ultra-semi-norm on E, then the sets W(p, $\alpha$ ) and V(p, $\alpha$ ) are both open and closed. For, we have seen that W(p, $\alpha$ ) is open; on the other hand if z is a cluster point of W(p, $\alpha$ ), then there exists $y \in W(p, \alpha)$ such that $p(y - z) < \alpha$ , and from (SN $_{II}'$ ) we have $p(z) < \alpha$ , thus W(p, $\alpha$ ) is closed. Also, V(p, $\alpha$ ) is closed since p is continuous; further if $p(x) \leq \alpha$ and $p(y) \leq \alpha$ , then $p(x + y) \leq \alpha$ by (SN $_{II}'$ ), and this shows that V(p, $\alpha$ ) is open.

2. Topologies defined by semi-norms

Let p be a semi-norm on the vector space E over K; for every $\alpha > 0$ let V(p, $\alpha$ ) be the subset of those x of E for which $p(x) \leq \alpha$ . Clearly, if $x \in V(p, \alpha)$ and $\lambda \in K$ is such that $|\lambda| \leq 1$ , then $\lambda x \in V(p, \alpha)$ , in other words V(p, $\alpha$ ) is balanced. Further, for every $x_0 \in E$ , there exists a non-zero scalar $\mu \in K$ such that $|\mu| \geq p(x_0) \alpha^{-1}$ , therefore $\mu^{-1} x_0 \in V(p, \alpha)$ that is to say V(p, $\alpha$ ) is absorbent. Finally, from (SN $_{II}$ ), we have $V(p, \alpha/2) + V(p, \alpha/2) \subset V(p, \alpha)$ , and from (SN $_I$ ) that for every non-zero scalar $\lambda$ in K we have $\lambda V(p, \alpha) = V(p, |\lambda| \alpha)$ . We conclude from these remarks, by I, p. 7, prop. 4, that, when $\alpha$ varies in the set of numbers $> 0$ (or only in a sequence of strictly positive numbers tending to 0) then the sets V(p, $\alpha$ ) constitute a fundamental system of neighbourhoods of 0 for a topology compatible with the vector space structure of E; we say that this topology is defined by the semi-norm p. A vector space E with such a topology is called a semi-normed space. Note that if W(p, $\alpha$ ) is the subset of x of E such that $p(x) < \alpha$ , then the W(p, $\alpha$ ) constitute (where $\alpha > 0$ , or $\alpha$ varies in a strictly positive sequence of numbers tending to zero) a fundamental system of neighbourhoods of 0 for the topology defined by $p$ .

If $\Gamma$ is a set of semi-norms on $E$ , then the upper bound of the topologies defined by the semi-norms $p \in \Gamma$ is compatible with the vector space structure (I, p. 10, cor. 4). A fundamental system of neighbourhoods of 0, for this topology, is given by the finite intersections $\bigcap_i V(p_i, \alpha_i)$ where $p_i \in \Gamma$ and $\alpha_i > 0$ . This topology is said to be defined by the set of semi-norms $\Gamma$ . It is the coarsest topology on $E$ amongst those that are invariant under all translations and for which the semi-norms $p \in \Gamma$ are continuous.

Let $E$ be a topological vector space over $K$ : a system of semi-norms on $E$ , say $\Gamma$ , is called a fundamental system of semi-norms if the topology on $E$ is the same as the topology defined by $\Gamma$ .

Let $E$ be a vector space over $K$ , with the topology defined by a set of semi-norms $\Gamma$ . For every semi-norm $p$ , we have $p(x - z) \leq p(x - y) + p(y - z)$ , which shows that the function $(x, y) \mapsto p(x - y)$ is a pseudometric on $E$ (GT, IX, § 1.1): it follows from the definitions that, when $p$ varies in $\Gamma$ , the set of these pseudometrics defines the uniform structure of the topological vector space $E$ .

Remark

— 1) The topology defined by a finite set of semi-norms $p_i$ ( $1 \leq i \leq n$ ) on $E$ , can be defined by the single semi-norm $p = \sup_{1 \leq i \leq n} p_i$ . But a topology defined by an infinite set of semi-norms cannot, in general, be defined by a single semi-norm (III, p. 37, exerc. 2).

2) Let $(\mathscr{T}_i)_{i \in I}$ be a family of topologies on a vector space $E$ over $K$ , each of which is defined by a family of semi-norms $\Gamma_i$ . Then the topology defined by the set of semi-norms $\Gamma = \bigcup_{i \in I} \Gamma_i$ is the upper bound of the topologies $\mathscr{T}_i$ .

3) If $\Gamma_0$ is a set of semi-norms directed by the increasing order relation defined between two semi-norms $p, q$ on $E$ by « there exists $\lambda > 0$ such that $p \leq \lambda q$ », then a fundamental system of neighbourhoods of 0, for the topology defined by $\Gamma_0$ , is obtained by taking the sets $V(p, \alpha)$ where $p \in \Gamma_0$ and $\alpha > 0$ . If $\Gamma$ is any set of semi-norms on $E$ , then a filtered set of semi-norms, defining the same topology as $\Gamma$ , is the set $\Gamma_0$ of upper envelopes of all finite families of semi-norms belonging to $\Gamma$ .

4) Even if $K = \mathbf{R}$ , the topology of a topological vector space over $K$ cannot always be defined by a set of semi-norms (cf. II, p. 24, corollary).

Example

— Let $\mathscr{C}^\infty(\mathbf{R})$ be the vector space over $\mathbf{R}$ of real valued functions that are infinitely differentiable in $\mathbf{R}$ . For every function and every pair of integers $n \geq 0$ , $m \geq 1$ , put

p_{n,m}(f) = \sup_{-m \leq t \leq m} |f^{(n)}(t)|

with $f^{(0)} = f$ . Obviously the $p_{n,m}$ are semi-norms on $\mathscr{C}^\infty(\mathbf{R})$ . In order that the functions $f_\alpha$ tend to 0 (following a filter $\mathfrak{F}$ on the set of indices) in $\mathscr{C}^\infty(\mathbf{R})$ for the topology $\mathscr{T}$ defined by the semi-norms $p_{n,m}$ , it is necessary and sufficient that for all integers $n \geq 0$ , the functions $f_\alpha^{(n)}$ tend to 0 (following $\mathfrak{F}$ ) uniformly on every compact subset of $\mathbf{R}$ . We say that $\mathscr{T}$ is the topology of compact convergence for the functions $f \in \mathscr{C}^\infty(\mathbf{R})$ and all their derivatives (cf. III, p. 9).

Proposition 2

On a vector space* $E$ , let $\mathscr{T}$ be the topology defined by a set of semi-norms $\Gamma$ .

(i) *The closure of {0} in E, for $\mathscr{T}$ , is the subset of $x \in E$ for which $p(x) = 0$ for every semi-norm $p \in \Gamma$ .

(ii) If $\mathscr{T}$ is Hausdorff and $\Gamma$ is enumerable, then $\mathscr{T}$ is metrisable.

The proposition follows immediately from the definitions and from GT, IX, § 2 . 4, cor. 1.

Note that if $\mathscr{T}$ is metrisable, it may be that $\mathscr{T}$ cannot be defined by a single norm; this is the case in the example given above (cf. IV, p. 18, Example 4).

Let E be a vector space over K, with the topology defined by a set of semi-norms $\Gamma$ . Let $\hat{E}$ be the Hausdorff completion of E (I, p. 6), and $\hat{\Gamma}$ be the set of mappings $\hat{p}$ of $\hat{E}$ in $\mathbf{R}_+$ where $p$ varies in $\Gamma$ (GT, II, § 3 . 7, prop. 15). By the principle of extending inequalities, the functions $\hat{p} \in \hat{\Gamma}$ are semi-norms on $\hat{E}$ , and the functions $\hat{p}(x - y)$ form a set of pseudometrics defining the uniform structure of $\hat{E}$ (GT, IX, § 1 . 3, prop. 1). We see, therefore, that $\hat{\Gamma}$ is a fundamental set of semi-norms defining the topology of $\hat{E}$ .

3. Semi-norms in quotient spaces and in product spaces

Let E be a topological vector space over K, whose topology is defined by $\Gamma$ , a set of semi-norms. Clearly, the restrictions of the semi-norms of $\Gamma$ to a vector sub-space M of E, define the topology induced on M by that of E.

Let $\phi$ be the canonical mapping of E on the vector quotient space E/M. We show that, for every semi-norm $p$ on E, the function

\dot{p}(z) = \inf_{\phi(x) = z} p(x)

is a semi-norm on E/M. In fact, it is clear that $\dot{p}$ satisfies the condition (SN $_1$ ); on the other hand, if $z', z''$ are two vectors of E/M, we have:

\begin{align*} \inf_{\phi(x) = z' + z''} p(x) &\leq \inf_{\phi(x') = z', \phi(x'') = z''} p(x' + x'') \\ &\leq \inf_{\phi(x') = z', \phi(x'') = z''} (p(x') + p(x'')) \\ &= \inf_{\phi(x') = z'} p(x') + \inf_{\phi(x'') = z''} p(x'') \end{align*}

which shows that $\dot{p}$ verifies (SN $_2$ ). We say that $\dot{p}$ is the quotient semi-norm of $p$ by M.

The same reasoning proves that, if $p$ is an ultra-semi-norm, then so also is $\dot{p}$ .

This being so, we have (in the notation of No. 2)

\phi(\mathrm{W}(p, \alpha)) = \mathrm{W}(\dot{p}, \alpha)

for every $\alpha > 0$ . In fact, to say that $\dot{p}(z) < \alpha$ , means that there exists $x \in E$ such that $\phi(x) = z$ and $p(x) < \alpha$ , from which the relation (4) follows.

We deduce from this, that, if the set of semi-norms $\Gamma$ is directed (II, p. 3, Remark 3), then the quotient topology on $E/M$ is defined by the set of semi-norms $\dot{p}$ , when $p$ varies in $\Gamma$ .

If $N$ is the closure of $\{0\}$ in $E$ , the topology of $E/N$ is defined by the quotient semi-norms $\dot{p}$ , where $p$ varies in $\Gamma$ (even if $\Gamma$ is not filtered): here $\dot{p}(\dot{x}) = p(x)$ for every $x$ belonging to the class $\dot{x} \mod N$ . Note that $E/N$ is none other than the Hausdorff space associated with $E$ (I, p. 4).

Let $E$ be a vector space over $K$ and $(E_i)_{i \in I}$ be a family of vector spaces over $K$ , where $E_i$ has the topology $\mathscr{T}_i$ defined by a set of semi-norms $\Gamma_i$ . For each $i \in I$ , let $f_i$ be a linear mapping of $E$ in $E_i$ ; clearly when $p_i$ varies in the set $\Gamma_i$ , then the $p_i \circ f_i$ form a set $\Gamma'_i$ of semi-norms on $E$ . The topology $\mathscr{T}$ on $E$ , defined as being the coarsest of all those which make all the mappings $f_i$ continuous (I, p. 9) is then defined by the set of semi-norms $\Gamma' = \bigcup_{i \in I} \Gamma'_i$ , this follows from the definition of neighbourhoods of 0 for $\mathscr{T}$ (GT, I, § 2.3, prop. 4).

If the $p_i$ are ultra-semi-norms, then so are the $p_i \circ f_i$ .

Let $E$ be a vector space over $K$ , with the topology $\mathscr{T}$ defined by a family of semi-norms $(p_i)_{i \in I}$ ; for every $i \in I$ , let $\mathscr{T}_i$ be the topology defined by the single semi-norm $p_i$ , and denote by $E_i$ the space obtained from $E$ using the topology $\mathscr{T}_i$ . Then the topology $\mathscr{T}$ is the inverse image by the diagonal mapping $\Delta : E \to \prod_{i \in I} E_i$ of the product topology on $\prod_{i \in I} E_i$ (I, p. 9, prop. 7). For each $i \in I$ , write $N_i$ for the closure of $\{0\}$ in $E_i$ , and by $F_i = E_i / N_i$ , the normed space defined by the norm $p_i$ corresponding to $p_i$ (II, p. 4, formula (3)); if $\phi_i : E_i \to F_i$ is the canonical mapping and $\phi : (x_i) \mapsto (\phi_i(x_i))$ the product mapping, we know that the product topology on $\prod_{i \in I} E_i$ is the inverse image by $\phi$ of the product topology on $\prod_{i \in I} F_i$ (GT, II, § 3.9, prop. 18). The topology $\mathscr{T}$ is, therefore, the inverse image under the composite mapping $\phi \circ \Delta$ of the product topology on $\prod_{i \in I} F_i$ . In particular, if $\mathscr{T}$ is Hausdorff then it follows from II, p. 3, prop. 2 that the mapping $\phi \circ \Delta$ is injective, therefore:

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.

If, further, the topology of $E$ is defined by an enumerable set of semi-norms, then $E$ is metrisable (I, p. 16).

4. Equicontinuity criteria of multilinear mappings for topologies defined by semi-norms

Proposition 4

Let* $E_i$ ( $1 \leq i \leq n$ ) and $F$ be topological vector spaces over $K$ ; we suppose that, for every $i$ , the topology of $E_i$ is defined by a directed set of semi-norms $\Gamma_i$ , and that the topology of $F$ is defined by a set of semi-norms $\Gamma$ . Then a set $H$ , of multilinear mappings of $\prod_{i=1}^n E_i$ in F is equicontinuous if, and only if, for each semi-norm $q \in \Gamma$ , and each index i, there exists a semi-norm $p_i \in \Gamma_i$ , and a number $a > 0$ , such that for each function $u \in \mathbf{H}$ and point $(x_i) \in \prod_{i=1}^n E_i$ ,

q(u(x_1, x_2, ..., x_n)) \leq a \cdot p_1(x_1)\ p_2(x_2) ... p_n(x_n) .

The condition is sufficient since it implies that H is equicontinuous at (0, 0, ..., 0) and therefore everywhere (I, p. 9, prop. 6).

We show that the condition is necessary. By hypothesis, for every semi-norm $q \in \Gamma$ and every number $\beta > 0$ , we have $q(u(x_1, x_2, ..., x_n)) \leq \beta$ for every function $u \in \mathbf{H}$ provided that $p_i(x_i) \leq \alpha_i$ are true for each index $i$ , $1 \leq i \leq n$ , and certain appropriately chosen numbers $\alpha_i > 0$ and semi-norms $p_i \in \Gamma_i$ . As K is non-discrete, we can also suppose that, for every $i$ , we have $\alpha_i = |\lambda_i| < 1$ where $\lambda_i \in \mathbf{K}$ . Then let $(x_1, x_2, ..., x_n)$ be any point of $\prod_{i=1}^n E_i$ , and for each index $i$ , let $m_i \in \mathbf{Z}$ be an integer such that $p_i(x_i) \leq |\lambda_i|^{m_i + 1}$ ; this can be written as $p_i(\lambda_i^{-m_i} x_i) \leq |\lambda_i|$ ( $1 \leq i \leq n$ ), therefore, by hypothesis, we have

q(u(x_1, x_2, ..., x_n)) \leq \beta |\lambda_1|^{m_1} |\lambda_2|^{m_2} ... |\lambda_n|^{m_n} .

Suppose firstly that one of the $p_i(x_i)$ is zero, then we can take $m_i \in \mathbf{N}$ arbitrarily large, therefore $q(u(x_1, x_2, ..., x_n)) = 0$ . If, on the contrary, all the $p_i(x_i)$ are $\neq 0$ , take the integer $m_i$ such that $|\lambda_i|^{m_i + 2} < p_i(x_i) \leq |\lambda_i|^{m_i + 1}$ for each $i$ ; then we have $|\lambda_i|^{m_i} < |\lambda_i|^{-2} p_i(x_i)$ , from which, by (6), the relation (5) follows with

a = \beta (|\lambda_1| \cdot |\lambda_2| ... |\lambda_n|)^{-2} .

Q.E.D.

Corollary

— *The set H is equicontinuous if, and only if, for every semi-norm $q \in \Gamma$ , there exists a neighbourhood of 0 in $\prod_{i=1}^n E_i$ , in which the functions $q \circ u$ , for $u \in \mathbf{H}$ , are uniformly bounded.

The condition is evidently necessary, and the demonstration of prop. 4 shows that it implies an inequality of the form (5) for all $u \in \mathbf{H}$ , and therefore the equicontinuity of H.

We state explicitly the particular case of prop. 4 for linear mappings.

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 $\Gamma$ (resp. $\Gamma'$ ). Let H be a set of linear mappings of E in F. The following conditions are equivalent : a) H is equicontinuous.

b) For every semi-norm $q \in \Gamma'$ , there exists a finite family $(p_i)_{1 \leq i \leq n}$ of semi-norms belonging to $\Gamma$ and a number $a > 0$ such that, for all $x \in E$ and all $u \in H$ ,

q(u(x)) \leq a \cdot \sup_{1 \leq i \leq n} p_i(x) .

c) For every semi-norm $q \in \Gamma'$ , the mapping $\sup_{u \in H} (q \circ u)$ is a continuous semi-norm on $E$ .

Corollary 1

— Suppose that $\mathscr{T}, \mathscr{T}'$ are two topologies on a vector space $E$ over $K$ defined, respectively, by two sets of semi-norms $\Gamma$ and $\Gamma'$ . $\mathscr{T}$ is finer than $\mathscr{T}'$ if, and only if, for every semi-norm $q \in \Gamma'$ , there exists a finite family $(p_i)_{1 \leq i \leq n}$ of semi-norms belonging to $\Gamma$ and a number $a > 0$ such that, for all $x \in E$ , we have $q(x) \leq a \cdot \sup_{1 \leq i \leq n} p_i(x)$ .

In fact this shows that the identity mapping of $E$ with topology $\mathscr{T}$ , on $E$ with topology $\mathscr{T}'$ , is continuous.

Corollary 2

— Suppose that the topology $\mathscr{T}$ of a topological vector space $E$ over $K$ is defined by a directed set of semi-norms $\Gamma$ ; for each semi-norm $p \in \Gamma$ , let $E_p$ be the space obtained from $E$ using the topology defined by $p$ . The set $E'$ of linear forms on $E$ that are continuous for $\mathscr{T}$ is the union of the sets $E'_p$ , where $E'_p$ is the set of continuous linear forms in $E_p$ ( $p \in \Gamma$ ).