UDC 513.83

MATHEMATICS

P. KENDEROV

DUALITY THEORY FOR TOPOLOGICAL VECTOR SPACES OVER A DISCRETE FIELD

(Presented by Academician P. S. Aleksandrov on 18 XI 1968)

I. Definition 1. A linear space \(E\) over a field \(P\), endowed with a topology \(u\), is called a topological vector group (\(^1\)) (abbreviated t.v.g.) if the mappings

\[ \xi_1:E\times E\to E\bigl(\xi_1(x,y)=x+y\bigr),\qquad \xi_2:P\times E\to E\bigl(\xi_2(\lambda,x)=\lambda x\bigr) \]

are continuous when \(P\) is endowed with the discrete topology. \(E\), endowed with the discrete topology, is a t.v.g. Every topological linear space is also a t.v.g. Linear topological spaces are also t.v. groups ((\(^2\)), § 10).

Everywhere in what follows it is assumed that \(P\) is either the field of real numbers \(R\) or the field of complex numbers \(C\).

We shall consider only those t.v.g.’s which possess a base of neighborhoods of zero consisting of convex sets. Such t.v.g.’s are called locally convex groups (l.c.g.).

L.c.g.’s possess a base of neighborhoods of zero \(\eta\), which satisfies the conditions: a) every \(U\in\eta\) is absolutely convex; b) for any \(\lambda\in P\), \(\lambda\ne 0\), and \(U\in\eta\), there exists \(V\in\eta\) such that \(V\subset \lambda U\).

For \(P=R\) this assertion is obvious, and for \(P=C\) it was proved in (\(^1\)). Conversely, every filter base \(\eta\) in \(E\) possessing properties a) and b) is a base of neighborhoods of zero for some topology relative to which \(E\) becomes an l.c.g.

The conjugate \(E'\equiv(E,u)'\) to the l.c.g. \((E,u)\) is the set of all continuous linear mappings \(f:E\to P\), where \(P\) is endowed with the topology usual for the line or the plane.

Theorem 1 (Hahn—Banach). Let \((E,u)\) be an l.c.g. and let \(f\) be a continuous functional defined on a linear subspace \(E_1\subset E\). Then there exists \(\bar f\in E'\) such that \(\bar f(x)=f(x)\) for \(x\in E_1\).

Corollary 1. If \((E,u)\) is a separated l.c.g. and \(x_0\in E\), \(x_0\ne 0\), then there exists \(f_0\in E'\) such that \(f_0(x_0)=1\).

By \(\sigma(E,E')\) we shall denote the weakest topology in \(E\) with respect to which all \(f\in E'\) are continuous. Let \((E,u)\) be an l.c.g. and \(E_1\) a linear subspace of \(E\); the topology \(u\) induces on \(E_1\) some topology \(u_1\). Put \(E_1'=(E_1,u_1)'\).

Corollary 2. The topology \(\sigma(E,E')\) induces on \(E_1\) the topology \(\sigma(E_1,E_1')\).

Corollary 3. Let \((E,u)\) be an l.c.g. and let \(E_1\) be a closed linear subspace; then \(E_1\) is \(\sigma(E,E')\)-closed.

Proposition 1. Let \(U\) be an absolutely convex neighborhood of zero of the l.c.g. \((E,u)\). Then

\[ \frac{1}{2}\overline{U}^{\sigma(E,E')}\subset U. \]

The meaning of this proposition is that every l.c.g. has a base of neighborhoods consisting of weakly closed absolutely convex sets.

Definition 2. Let \((E,\tau)\) be a separated l.c.g. A set \(A\subset E\) is called precompact if \(\bar A^{\,\tau}\) is complete in the topology \(\tau\). In other words, \(A\) is precompact if every \(\tau\)-Cauchy net \(\{x_\alpha\}\), \(x_\alpha\in A\), converges in \(E\).

Let \((E,F)\) be a separated dual pair with bilinear function \(\langle e,f\rangle\). For \(A\subset E\) put

\[ A^{0}=\{f\in F:\ |\langle A,f\rangle|\leq 1\}. \]

Theorem 2 (V. L. Shmul′yan; see \((^3)\)). Let \((E,F)\) be a separated dual pair. An absolutely convex set \(A\subset E\) is \(\sigma(E,F)\)-precomplete if and only if every linear function \(s\) on \(F\) \((s\in F^*)\), restricted to \(A^0\), is representable by an element of \(E\) (i.e. \(s(f)=\langle e_s,f\rangle\) for some \(e_s\in E\)).

Theorem 3 (Banach—Alaoglu). Let \((E,\tau)\) be an l.v.s., and let \(U\) be a neighborhood of zero in \((E,\tau)\). Then \(U^0\) is a \(\sigma(E',E)\)-complete set in \(E'\).

Let \((E,F)\) be a separated dual pair and let \(\tau\) be an l.v.s. topology in \(E\), consistent with the duality between \(E\) and \(F\), i.e. \((E,\tau)'\) is isomorphic to \(F\). Let \(\{U_\gamma:\gamma\in\Gamma\}\) be a base of neighborhoods of zero for \((E,\tau)\), consisting of \(\sigma(E,F)\)-closed absolutely convex sets. Since \(U_\gamma^{00}=U_\gamma\), by Theorem 3, \(\tau\) is the topology of uniform convergence on the family \(\{U_\gamma^0:\gamma\in\Gamma\}\) of \(\sigma(F,E)\)-complete absolutely convex subsets of \(F^*\).

It turns out that the topology \(\nu(E,F)\) of uniform convergence on all absolutely convex \(\sigma(F,E)\)-complete subsets of \(F\) is also consistent with the duality between \(E\) and \(F\).

Theorem 4. Let \((E,F)\) be a separated dual pair. Every l.v.s. topology \(\tau\) in \(E\) consistent with the duality between \(E\) and \(F\) satisfies the condition
\[ \sigma(E,F)\leq \tau\leq \nu(E,F) \]
and is the topology of uniform convergence on some family of absolutely convex \(\sigma(F,E)\)-complete subsets of \(F\).

In this theorem the words “\(\sigma(E,F)\)-complete” may be replaced by the words “\(\sigma(F,E)\)-precomplete.”

Corollary 1. The absolutely convex hull \(\Gamma(A_1\cup A_2)\) of the union of two \(\sigma(F,E)\)-precomplete absolutely convex sets \(A_1\) and \(A_2\) is a \(\sigma(F,E)\)-precomplete set.

II. From Theorem 4 it follows

Theorem 5. Let \(\varphi:E\to F\) be a continuous linear mapping of a separated l.v.s. \(E\) into a separated l.v.s. \(F\), and let \(A\) be a \(\sigma(E,E')\)-precomplete absolutely convex subset of \(E\). Then \(\varphi A\) is a \(\sigma(F,F')\)-precomplete set and
\[ \varphi A^{\sigma(F,F')}\subset \varphi E. \]

From Theorem 5 and Theorem 3 it follows

Proposition 2. A continuous linear mapping \(\varphi:E\to F\) of a separated l.v.s. \(E\) into a separated l.v.s. \(F\) remains continuous if \(E\) and \(F\) are endowed with the topologies \(\nu(E,E')\) and \(\nu(F,F')\), respectively.

Corollary. The quotient group \(F\) of an l.v.s. \(E\), endowed with the topology \(\nu(E,E')\), is an l.v.s. with the topology \(\nu(F,F')\).

III. Let \(E\) be a separated l.v.g. with adjoint \(E'\). A set \(A'\subset E'\) will be called, following V. Pták \((^5)\), almost closed if \(A'\cap U^0\) is \(\sigma(E',E)\)-closed for every neighborhood \(U\) in \(E\). An l.v.g. \(E\) will be called perfectly complete if every almost closed linear subspace in \(E'\) is \(\sigma(E',E)\)-closed.

A linear mapping \(\varphi:(E,u)\to(F,v)\) of a separated l.v.g. \((E,u)\) into a separated l.v.g. \((F,v)\) will be called almost open if \(\varphi U^{\sigma(F,F')}\) is a neighborhood of zero in \((F,v)\) whenever \(U\) is a neighborhood of zero in \((E,u)\)**. Analogously one defines almost continuity of a mapping \(\varphi:(E,u)\to(F,v)\).

* This assertion, as well as Proposition 1, was obtained independently of the author by A. Merzon.

** The closure of the set \(\varphi U\) with respect to the topology \(\sigma(F,F')\) may be larger than the closure of the same set in the topology \(v\), even when \(U\) is an absolutely convex set.

Theorem 6. Let \(E\) and \(F\) be separated l.v.s., and let \(\varphi:E\to F\) be a linear mapping with closed graph.

a) If \(\varphi\) is almost continuous, and \(F\) is a perfectly complete l.v.s., then \(\varphi\) is continuous.

a) If \(\varphi\) is almost open, \(\varphi E=F\), and \(E\) is a perfectly complete l.v.s., then \(\varphi\) is open.

Theorem 6 is proved with the aid of Theorem 5 according to the scheme proposed in \({}^{4}\), Ch. VI, § 2.

IV. Let \(E\) be a separated l.v.s. We shall say \({}^{6}\) that a set \(A\subset E\) satisfies the double-limit condition if, for every neighborhood of zero \(U\) in \(E\), and for any two sequences \(\{e_i\}_{i=1}^{\infty}\subset A\) and \(\{f_j\}_{j=1}^{\infty}\subset U^0\), from the existence of the double limits \(\lim_i\lim_j\langle e_i,f_j\rangle\) and \(\lim_j\lim_i\langle e_i,f_j\rangle\) it follows that they are equal.

Proposition 3. Let \(E\) be a complete separated l.v.s., and let \(A\subset E\) satisfy the double-limit condition. Then \(A\) is \(\sigma(E,E')\)-precomplete.

Theorem 7 (Eberlein). Let \(E\) be a separated complete l.v.s., and let \(A\) be a subset such that every sequence of it has a weak cluster point. Then \(\overline{A}^{\sigma(E,E')}\) is weakly compact.

With the aid of Theorem 5 it is proved that \(A\) satisfies the double-limit condition.

Proposition 4. Let \(B\) be a \(\sigma(E,E')\)-precomplete absolutely convex subset of a separated l.v.s. \(E\). Then \(B\) satisfies the double-limit condition.

This assertion is also proved by applying Theorem 5.

From Propositions 3 and 4 it follows:

Theorem 8. An absolutely convex subset \(A\) of a complete separated l.v.s. \(E\) is \(\sigma(E,E')\)-precomplete if and only if it satisfies the double-limit condition.

The work was carried out under the supervision of Prof. D. A. Raikov, to whom the author expresses his sincere gratitude.

References

\({}^{1}\) D. A. Raikov, Studia Math., 30, 295 (1968).
\({}^{2}\) G. Köthe, Topologische lineare Räume, Berlin, 1960.
\({}^{3}\) D. A. Raikov, DAN, 111, No. 4, 760 (1956).
\({}^{4}\) A. P. Robertson, W. J. Robertson, Topological Vector Spaces, Moscow, 1967.
\({}^{5}\) V. Pták, Czechoslovak Math. Journal, 3 (78), No. 4, 301 (1953).
\({}^{6}\) A. Grothendieck, Am. J. Math., 74, 168 (1952).