Clearly $\mathfrak{B}$ is a filter base satisfying (EV_I) and (EV_{II}) of I, p. 7, prop. 4; it also satisfies (EV_{III}) since if $V \in \mathfrak{S}$ then $\frac{1}{2}V + \frac{1}{2}V = V$.

Note that if $\mathscr{T}$ is the locally convex topology on E having $\mathfrak{B}$ for a fundamental system of neighbourhoods of 0, then the sets $(1/n) \ V$, where $n$ varies in the integers $> 0$ and V varies in $\mathfrak{S}$, form a fundamental system of neighbourhoods of 0 for the topology $\mathscr{T}$. Then $\mathscr{T}$ is Hausdorff, if and only if, for every $x \neq 0$ in E there exists an integer $n$ and a set $V \in \mathfrak{S}$, such that $nx \notin V$; if, further, $\mathfrak{S}$ is enumerable, then the topology $\mathscr{T}$ is a metrisable locally convex topology. Conversely, it is clear that if $\mathscr{T}$ is a metrisable locally convex topology, then there exists an enumerable fundamental system of closed symmetric convex neighbourhoods of 0 for $\mathscr{T}$.

The condition is necessary since every semi-norm on E is a convex function, and so, for $\alpha > 0$, the set of $x \in E$ for which $p(x) \leq \alpha$, is convex (II, p. 17, corollary). Conversely if V is a symmetric, closed, convex neighbourhood of 0 in E, the gauge $p$ of V is a semi-norm on E such that V is the set of points $x$ of E satisfying $p(x) \leq 1$ (II, p. 20, prop. 23).

This shows further that a locally convex topology $\mathscr{T}$ is defined by the set of all semi-norms that are continuous for $\mathscr{T}$. Further, if $\mathscr{T}$ is metrisable, then it is defined by an enumerable set of semi-norms.

From the corollary to prop. 1, all the results of § 1 on topologies defined by sets of semi-norms apply in particular to locally convex topologies over real vector spaces. A locally convex Hausdorff space E has a completion $\hat{E}$ that is locally convex. A complete, metrisable locally convex space is called a Fréchet space; every Banach space is a Fréchet space.

From the corollary above and II, p. 7, cor. 2, there exists a continuous semi-norm $p$ on E, such that $|f(y)| \leq p(y)$ for all $y \in M$. By the Hahn-Banach th. (II, p. 23, cor. 1) there exists a linear form $h$ on E that extends $f$ and is such that $|h(x)| \leq p(x)$ for all $x \in E$, and this implies that $h$ is continuous (II, p. 6, prop. 5).

Apply prop. 2 to the one dimensional vector space M generated by $x_0$ and to the linear form $\xi x_0 \mapsto \xi$ defined over M, which, by I, p. 12, prop. 2, is continuous.

There exists a topological complement to M in E if, and only if, the identity mapping of M on itself can be extended to a continuous linear mapping of E on M, which mapping is then necessarily a continuous projector (GT, III, § 6.2, corollary). Now, this follows from the remark above since M is isomorphic to a space $\mathbf{R}^n$ (I, p. 13, th. 2).

Let A be a precompact set in E. Given V, a balanced convex neighbourhood of 0 in E, there exist finitely many points $a_i \in A$ ($1 \leq i \leq n$) such that A is contained in S, the union of the neighbourhoods $a_i + V$ ($1 \leq i \leq n$). Thus C, the balanced convex envelope of A, is contained in T the balanced convex envelope of S; but T is contained in B + V, where B denotes the convex envelope of the finite set of points $a_i, -a_i$ ($1 \leq i \leq n$). Now B is precompact (II, p. 14, cor. 2); hence there exist finitely many points $b_k \in B$ ($1 \leq k \leq m$) such that $B_k$ is contained in the union of the neighbourhoods $b_k + V$. Then C is contained in the union of the neighbourhoods $b_k + 2V$, and the proposition is proved.

Note that, in an infinite dimensional locally convex Hausdorff space, the convex envelope of a compact set is not necessarily closed (II, p. 74, exerc. 3).

For this envelope is a closed subset of a complete space, therefore it is complete, but it is also precompact and Hausdorff.

However in a non complete locally convex Hausdorff space, the convex closed envelope of a compact set need not be compact (II, p. 87, exerc. 2).

Using II, p. 24, corollary, this is a particular case of the corresponding property for topologies defined by semi-norms (II, p. 5).

In particular, every vector subspace of a locally convex space, and every product space of locally convex spaces, is locally convex. Every projective limit of locally convex spaces is locally convex.

Every enumerable product of Fréchet spaces (and in particular every enumerable product of Banach spaces) is a Fréchet space.

Every locally convex Hausdorff space E is isomorphic to a subspace of a product of Banach spaces and this subspace is closed if E is complete (II, p. 5, prop. 3). Every Fréchet space is isomorphic to a closed subspace of an enumerable product of Banach spaces (loc. cit.).

(i) Denote by $\mathfrak{B}$ the family of absorbent, symmetric convex subsets V of E such that $g_\alpha^{-1}(V)$ is a neighbourhood of 0 in $F_\alpha$ for every $\alpha$; the family $\mathfrak{B}$ is a fundamental system of neighbourhoods of 0 in E for a topology $\mathscr{T}$ that is compatible with the vector space structure.

(ii) A linear mapping f of E in a locally convex space G (resp. a semi-norm p on E) is continuous for $\mathscr{T}$ if and only if, for every index $\alpha$, $f \circ g_\alpha$ (resp. $p \circ g_\alpha$) is continuous in $F_\alpha$.

(iii) The topology $\mathscr{T}$ is the finest of the locally convex topologies on E for which the $g_\alpha$ are continuous.

Further, the topology $\mathscr{T}$ is the only locally convex topology on E that satisfies condition (ii) for linear mappings (resp. for the semi-norms).

As $\mathfrak{B}$ is a filter base invariant under homotheties of ratio > 0, the assertion (i) follows immediately from II, p. 23, prop. 1. By the definition of $\mathfrak{B}$, the topology $\mathscr{T}$ is the finest of locally convex topologies on E making the $g_\alpha$ continuous; whence (iii). Finally, it is clear that if $f$ is continuous, so is $f \circ g_\alpha$; conversely if the $f \circ g_\alpha$ are continuous for every $\alpha$, then for each symmetric convex neighbourhood W of 0 in G, the set $g_\alpha^{-1}(f^{-1}(W))$ is a neighbourhood of 0 in $F_\alpha$ for each $\alpha$. Now $f^{-1}(W)$ is absorbent, symmetric and convex thus $f^{-1}(W)$ is a neighbourhood of 0 in $\mathscr{T}$, and $f$ is continuous. Similarly if $p$ is a semi-norm on E such that $p \circ g_\alpha$ is continuous for every $\alpha$, and if U is the set of points $x \in E$ such that $p(x) < 1$, then, for every $\alpha$, the set $g_\alpha^{-1}(U)$ is a convex neighbourhood of 0 in $E_\alpha$ that is symmetric and absorbent; thus U is a neighbourhood of 0 in E and $p$ is continuous (II, p. 2, prop. 1).

The last statement follows from S, IV, § 2.5, criterion CST 18.

We say that $\mathscr{T}$ is the locally convex final topology of the family of topologies $\mathscr{T}_\alpha$ of the $F_\alpha$, for the family of linear mappings $g_\alpha$.

It may happen that $\mathscr{T}$ is not the finest of the topologies on E compatible with its vector space structure and making the $f_\alpha$ continuous (II, p. 75, exerc. 15; see also II, p. 75, exerc. 14).

In the most important case $E = \sum_{\alpha \in A} g_\alpha(F_\alpha)$, we get a fundamental system of neighbourhoods of 0 for $\mathscr{T}$ as follows; for each $\alpha \in A$, let $V_\alpha$ be a symmetric neighbourhood of 0 for $\mathscr{T}_\alpha$, form the union of the $g_\alpha(V_\alpha)$ for $\alpha \in A$ and denote the convex envelope in E of this union by $\Gamma((g_\alpha(V_\alpha)))$; since every element of E is of the form $\sum_{\alpha \in J} x_\alpha$, where J is a finite subset of I and $x_\alpha \in g_\alpha(F_\alpha)$, it is immediate that $\Gamma((g_\alpha(V_\alpha)))$ is an absorbent symmetric convex set in E (each of the $V_\alpha$ is absorbent in $F_\alpha$); as $\Gamma((g_\alpha(V_\alpha)))$ contains all the $g_\alpha(V_\alpha)$, it is a neighbourhood of 0 for $\mathscr{T}$. On the other hand, it is clear that for every symmetric convex neighbourhood V of 0 for $\mathscr{T}$, we have $V \supset \Gamma((V \cap g_\alpha(F_\alpha)))$, from which our assertion follows.

Remembering I, p. 9, prop. 6 the argument is similar to that of (ii) prop. 5. Let W be a symmetric convex neighbourhood of 0 in G and note that if the set $f \circ g_\alpha$, where $f \in H$ is equicontinuous, then the intersection $\bigcap_{f \in H} g_\alpha^{-1}(f^{-1}(W))$ is a symmetric convex neighbourhood of 0 in $F_\alpha$. As this intersection is the same as $g_\alpha^{-1}(\bigcap_{f \in H} f^{-1}(W))$ and the set $\bigcap_{f \in H} f^{-1}(W)$ is symmetric and convex, everything depends on showing that it is also absorbent. Now, by hypothesis, every $x \in E$ can be written as $\sum_{i=1}^n g_{\alpha_i}(z_{\alpha_i})$, where $z_{\alpha_i} \in F_{\alpha_i}$. To show that there exists $\lambda > 0$ such that $f(\lambda x) \in W$ for all $f \in H$, it is sufficient to consider the case $x = g_\alpha(z_\alpha)$ with $z_\alpha \in F_\alpha$ (since we can pass to the general case by replacing W by $W/n$). But this case follows from the fact that $g_\alpha^{-1}(\bigcap_{f \in H} f^{-1}(W))$ is a neighbourhood of 0 in $F_\alpha$.

This is a particular case of S, IV, § 2.5 criterion CST 19, and can also be proved directly using prop. 5.

Examples of locally convex final topologies.

I. Quotient space.

Let M be a subspace of the locally convex space F, and φ be the canonical mapping of F on F/M. As the quotient topology on F/M is locally convex and is the finest of all the topologies (locally convex or not) which make φ continuous, it is also the locally convex final topology for the family consisting of the single mapping φ.

II. Inductive limits of locally convex spaces.

Let A be an ordered set directed to the right and let $(E_\alpha, f_{\beta\alpha})$ be an inductive system of vector spaces relative to the set A (A, II, § 6.2); let $E = \varinjlim E_\alpha$ and let $f_\alpha : E_\alpha \to E$ be the canonical linear mapping for each $\alpha \in A$. Suppose that each $E_\alpha$ carries a locally convex topology $\mathscr{T}_\alpha$, and further suppose that for $\alpha \leq \beta$, the mapping $f_{\beta\alpha} : E_\alpha \to E_\beta$ is continuous. Then we say that the locally convex final topology $\mathscr{T}$ of the family $(\mathscr{T}_\alpha)$ relative to the linear mappings $f_\alpha$ (resp. the space E carrying the topology $\mathscr{T}$) is the inductive limit of the family $(\mathscr{T}_\alpha)$ (resp. the inductive limit space of the system $(E_\alpha, f_{\beta\alpha})$, or simply of the locally convex spaces $E_\alpha$). Recall that E is the union of the vector subspaces $f_\alpha(E_\alpha)$ and that when $\alpha \leq \beta$, we have $f_\alpha(E_\alpha) \subset f_\beta(E_\beta)$; if we endow $f_\alpha(E_\alpha)$ with the final topology for the mapping $f_\alpha$ (which is the same as identifying $f_\alpha(E_\alpha)$ with the quotient space $E_\alpha/f_\alpha^{-1}(0)$), the topology $\mathscr{T}$ is also the final topology of the family of the topologies of the $f_\alpha(E_\alpha)$, relative to the canonical injections (II, cor. 2 above). Further, the continuity of $f_{\beta\alpha}$ for $\alpha \leq \beta$ implies that the canonical injection $j_{\beta\alpha} : f_\alpha(E_\alpha) \to f_\beta(E_\beta)$ is continuous, so that E is also the inductive limit of $f_\alpha(E_\alpha)$ carrying the preceding topologies relative to the injection $j_{\beta\alpha}$.

for every locally convex space $G$.

This is a particular case of II, p. 28, cor. 2 remembering II, p. 29, Example I.

The first assertion follows from the fact that the canonical injections $E_\kappa \to \prod_{\iota \in I} E_\iota$ are continuous for each $\kappa \in I$. If $I$ is finite then $j$ is the identity mapping, and it is sufficient to show that the product topology $\mathscr{T}'$ is finer than the direct sum topology $\mathscr{T}$. Now, let $V$ be a convex neighbourhood of 0 for $\mathscr{T}$; each set $V \cap E_\iota$ is a convex neighbourhood of 0 in $E_\iota$; if $n$ is the number of elements of $I$, then the set $V$ contains the set $\frac{1}{n} \sum_n (V \cap E_\iota)$, which is a neighbourhood of 0 for $\mathscr{T}'$, and the proposition is proved.

When $I$ is infinite, if, for each finite subset $J$ of $I$, we write $E_J$ for the space $\prod_{\iota \in J} E_\iota$, with the product topology, then $E$ is the inductive limit of the $E_J$ (identified as subspaces of $E$).

(i) The topology induced on $N = \sum_\iota N_\iota$ by the direct sum topology $\mathscr{T}$ on $E$ is identical with the direct sum of the topologies of the $N_\iota$.

(ii) The canonical mapping $h$ of the topological direct sum space of the $E_\iota / N_\iota$ on $E/N$ (A, II, § 1.6, formula (26)) is an isomorphism between topological vector spaces.

(i) With the notations introduced above, we consider $x = \sum_\iota \lambda_\iota x_\iota$ belonging to $N \cap \Gamma((V_\iota))$ ($(\lambda_\iota)$ is a family of numbers $\geqslant 0$ of which at most finitely many are non-zero, such that $\sum_\iota \lambda_\iota = 1$, and $x_\iota \in V_\iota$, for all $\iota \in I$). Since the sum of the $N_\iota$ is direct, we have $\lambda_\iota x_\iota \in N_\iota$ for all $\iota \in I$; therefore, for all $\iota$ such that $\lambda_\iota > 0$ we also have $x_\iota \in N_\iota \cap V_\iota$, and $x$ belongs to the convex envelope $\Gamma((N_\iota \cap V_\iota))$, thus (i) is proved.

(ii) Denote canonical mappings as follows: $f_\iota : E_\iota \to E, h_\iota : E_\iota / N_\iota \to E/N, p_\iota : E_\iota \to E_\iota / N_\iota$ and $p : E \to E/N$. For every $\iota \in I$, $h_\iota \circ p_\iota = p \circ f_\iota$ and the proposition follows from II, p. 28, cor. 2 and p. 29 Example I.

For, the canonical mapping $p_\iota : E \to E_\iota$ is continuous (II, § 4.5, prop. 6) for every $\iota \in I$, hence $p_\iota^{-1}(N_\iota)$ is closed in $E$, and thus the same is true of the intersection $N = \bigcap_{\iota \in I} p_\iota^{-1}(N_\iota)$.

To prove the first statement apply cor. 1 taking $N_\iota = \{0\}$ for all $\iota \in I$; for the second apply cor. 1 with $N_\iota = E_\iota$ and $N_\kappa = \{0\}$ for every $\kappa \neq \iota$.

We shall show in III, p. 21, cor. 2 that if the $E_\iota$ are Hausdorff and complete then so is their topological direct sum $E$.

We assume the contrary and obtain a contradiction. Suppose that for every $k$ there exists a convex neighbourhood $V_k$ of $0$ and a set $M_k \in \mathfrak{F}$ such that

Clearly we can suppose that $V_{k+1} \subset V_k$ for all $k$. Let $V$ be the convex envelope of $\bigcup_k (E_k \cap V_k)$, this is clearly a neighbourhood of $0$ for $\mathscr{T}$. For all $n$ we have $V \subset V_n + E_n$; in fact, every $x \in V$ can be written $\sum_i \lambda_i x_i$ where $\lambda_i \geq 0$, $\sum_i \lambda_i = 1$ and $x_i \in V_i \cap E_i$ for all $i$; now for $i < n$ we have $x_i \in E_n$, therefore $\sum_{i < n} \lambda_i x_i \in E_n$; and for $i \geq n$ we have $x_i \in V_n$, therefore $\sum_{i \geq n} \lambda_i x_i \in V_n$ since $V_n$ is convex, contains $0$ and $\sum_{i \geq n} \lambda_i \leq 1$. Hence $V + E_n \subset V_n + E_n$ for all $n$. This being so, let $M \in \mathfrak{F}$ be a set that is $V$-small. For some integer $m$, $E_m \cap M$ is not empty; and we conclude that as $\mathfrak{F}$ is a filter, the set $M_m$ meets $M$ and therefore $E_m + V_m$; we have a contradiction which establishes the lemma.

(i) To prove the first assertion, it is sufficient to prove that the topology $\mathscr{T}'_n$ induced by $\mathscr{T}$ on $E_n$ is finer than $\mathscr{T}_n$. For this, let $V_n$ be a convex neighbourhood of $0$ in $E_n$ for the topology $\mathscr{T}_n$; we are going to construct an increasing sequence of convex neighbourhoods of $0$ in $E_{n+p}$ for $\mathscr{T}_{n+p}$, say $(V_{n+p})_{p \geq 1}$, such that $V_{n+p} \cap E_n = V_n$ for every index $p \geq 1$. Then the union $V$ of the increasing sequence $(V_{n+p})$ will be a convex set such that $V \cap E_k$ is a neighbourhood of $0$ in $E_k$ (using $\mathscr{T}_k$), for every index $k$; therefore $V$ will be a neighbourhood of $0$ in $E$ for $\mathscr{T}$ and as $V \cap E_n = V_n$, we have proved that $\mathscr{T}'_n$ is finer than $\mathscr{T}_n$.

To define the $V_{n+p}$ we proceed by induction on $p$ using the following lemma:

In fact, by hypothesis there exists a convex neighbourhood $U$ of 0 in $F$ such that $U \cap M \subset V$. Clearly, the convex envelope $W$ of $U \cup V$ in $F$ is a neighbourhood of 0 in $F$; we show that $W \cap M = V$. For, every point $z \in W$ is of the form $\lambda x + (1-\lambda) y$ with $x \in V, y \in U$, and $0 \leq \lambda \leq 1$ (II, p. 9, prop. 8); if $z \in M$, and $\lambda \neq 1$ then necessarily $y \in M$, therefore $y \in U \cap M \subset V$ and hence $z \in V$; the result is obviously true if $\lambda = 1$. If $M$ is closed in $F$, the space $F/M$ is Hausdorff, thus there exists a convex neighbourhood $U_0 \subset U$ of 0 in $F$ such that $U_0$ does not meet $x_0 + M$; then the convex envelope $W_0$ of $U_0 \cup V$ fulfils the required conditions.

Returning to the theorem, to prove the second part of (i) note that if $x \in E$ then $x \in E_n$ for some $n$; if $x \neq 0$ and $\mathscr{T}_n$ is Hausdorff then there is a neighbourhood $V_n$ of 0 for $\mathscr{T}_n$, which does not contain $x$. We see that there is a neighbourhood $V$ of 0 for $\mathscr{T}$ such that $V \cap E_n = V_n$, hence $x \notin V$, and it follows that $\mathscr{T}$ is Hausdorff.

(ii) Let $x \in E - E_n$; there exists $m > n$ such that $x \in E_m$, thus, as $E_n$ is closed in $E_m$ for $\mathscr{T}_m$ (because of the hypothesis that $\mathscr{T}_{n+1}$ induces $\mathscr{T}_n$ on $E_n$ for every $n$) there exists in the topology $\mathscr{T}_m$ a convex neighbourhood $V_m$ of 0 in $E_m$ such that $(x + V_m) \cap E_n$ is empty. Now we saw in (i) that there exists a convex neighbourhood $V$ of 0 for $\mathscr{T}$ such that $V \cap E_m = V_m$; and thus $(x + V) \cap E_m = x + V_m$, therefore $(x + V) \cap E_n = \varnothing$, which proves (ii).

(iii) From lemma 1, if $\mathfrak{F}$ is a minimal Cauchy filter for $\mathscr{T}$ (GT, II, § 3.2) then there exists a $k$ such that the trace of $\mathfrak{F}$ on $E_k$ is a filter $\mathfrak{F}_k$; from (i) this last is a Cauchy filter for $\mathscr{T}_k$ and thus $\mathfrak{F}_k$ converges in $E_k$ by hypothesis; but as the filter on $E$ generated by $\mathfrak{F}_k$ is finer than $\mathfrak{F}$, we see that $\mathfrak{F}$ has a cluster point for $\mathscr{T}$ and thus converges for $\mathscr{T}$.

When for all $n$ the topology induced on $E_n$ by $\mathscr{T}_{n+1}$ is just $\mathscr{T}_n$ we say that $\mathscr{T}$ is the strict inductive limit of the sequence $(\mathscr{T}_n)$ and that the space $E$ with the topology $\mathscr{T}$ is the strict inductive limit of the sequence of locally convex spaces $E_n$.