UDC 513.88 : 513.83

MATHEMATICS

V. D. GOLOVIN

ON SOME EXTENSIONS OF TOPOLOGICAL VECTOR SPACES

(Presented by Academician S. N. Bernshtein, 21 VI 1966)

1. Let \(E\) be a separable locally convex topological vector space over the field \(R\) of real numbers, and let \(E'\) be the vector space conjugate to \(E\), endowed with the strong topology. Let \(A\) be an arbitrary nonempty set and let \(\mathfrak{B}(A)\) be the Banach algebra of all bounded real-valued numerical functions on \(A\). Define the \(\mathfrak{B}(A)\)-module

\[ E_{(A)} = L_R(E', \mathfrak{B}(A)) \]

of all continuous linear mappings of the topological vector space \(E'\) into the Banach space \(\mathfrak{B}(A)\). Considering the module \(E_{(A)}\) as a vector space over \(R\), define the linear mapping

\[ i: E \to E_{(A)}, \]

which assigns to each element \(s \in E\) the continuous linear mapping \(x: E' \to \mathfrak{B}(A)\) such that \(x(t)\), for every \(t \in E'\), is the function on \(A\) identically equal to \(t(s)\). It is immediately evident that the mapping \(i\) is injective; therefore \(E_{(A)}\) may be regarded as a module obtained from the topological vector space \(E\) by extending its field of scalars to the algebra \(\mathfrak{B}(A)\).

Analogously one may define the \(\mathfrak{B}(A)\)-module

\[ E'_{(A)} = L_R(E, \mathfrak{B}(A)) \]

of all continuous linear mappings \(E \to \mathfrak{B}(A)\) and the linear mapping

\[ j: E' \to E'_{(A)}, \]

which assigns to each element \(t \in E'\) the continuous linear mapping \(y: E \to \mathfrak{B}(A)\) such that \(y(s)\), for every \(s \in E\), is the function on \(A\) identically equal to \(t(s)\). Since the mapping \(j\) is injective, \(E'_{(A)}\) may be regarded as a module obtained from the space \(E'\) by extending its field of scalars to the algebra \(\mathfrak{B}(A)\).

1. Let \(\varepsilon_\alpha\), for each \(\alpha \in A\), be the continuous linear form on \(\mathfrak{B}(A)\) such that \(\varepsilon_\alpha(b) = b(\alpha)\) for every \(b \in \mathfrak{B}(A)\). Then each element \(x \in E_{(A)}\) is identified with the equicontinuous family of linear forms \(x_\alpha = \varepsilon_\alpha \circ x\) \((\alpha \in A)\) on the topological vector space \(E'\); analogously, each element \(y \in E'_{(A)}\) is identified with the equicontinuous family of linear forms \(y_\alpha = \varepsilon_\alpha \circ y\) \((\alpha \in A)\) on \(E\). Since every equicontinuous set in \(E'\) is bounded in the strong topology, the values \(\langle x_\alpha, y_\beta\rangle\) \((\alpha, \beta \in A)\) of the canonical bilinear form defined on the product \(E'' \times E'\) are bounded in the aggregate. We define on the product \(E_{(A)} \times E'_{(A)}\) a \(\mathfrak{B}(A)\)-bilinear form by assigning to each pair \((x,y) \in E_{(A)} \times E'_{(A)}\) the bounded numerical function \(\alpha \to \langle x_\alpha, y_\alpha\rangle\) on \(A\), which we shall denote by \(\langle x,y\rangle\). We shall show that the \(\mathfrak{B}(A)\)-bilinear form so defined,

\[ (x,y) \to \langle x,y\rangle \]

puts the modules \(E_{(A)}\) and \(E'_{(A)}\) in duality. Indeed—

therefore, if \(\langle x,y\rangle=0\) for some \(x\in \dot E_{(A)}\) and every \(y\in E'_{(A)}\), then \(\langle x_\alpha,t\rangle=0\) \((\alpha\in A;\ t\in E')\), whence \(x_\alpha=0\) \((\alpha\in A)\), i.e. \(x=0\). If \(\langle x,y\rangle=0\) for every \(x\in E_{(A)}\) and some \(y\in E'_{(A)}\), then \(\langle s,y_\alpha\rangle=0\) \((\alpha\in A;\ s\in E)\) and, consequently, \(y=0\). Finally, it is obvious that \(\varepsilon_\alpha(\langle i(s),j(t)\rangle)=\langle s,t\rangle\) \((\alpha\in A)\) for every pair \((s,t)\in E\times E'\). Thus the following has been proved.

Theorem 1. On the product \(E_{(A)}\times E'_{(A)}\) there exists a canonical \(\mathfrak B(A)\)-bilinear form \((x,y)\mapsto\langle x,y\rangle\), putting the modules \(E_{(A)}\) and \(E'_{(A)}\) in duality and such that the value \(\langle i(s),j(t)\rangle\) for each pair \((s,t)\in E\times E'\) identically coincides with the value \(\langle s,t\rangle\) of the canonical bilinear form putting the spaces \(E\) and \(E'\) in duality.

3. By the weak topology \(\sigma(E_{(A)},E'_{(A)})\) we shall mean the weakest of the topologies in \(E_{(A)}\) for which all \(\mathfrak B(A)\)-linear forms of the form \(x\mapsto\langle x,y\rangle\) \((y\in E'_{(A)})\) are continuous. This is a topology consistent with the structure of the \(\mathfrak B(A)\)-module in \(E_{(A)}\), for which a fundamental system of neighborhoods of zero is formed by the sets

\[ W(y_1,\ldots,y_n;\varepsilon)=\{x\in E_{(A)}:\ \|\langle x,y_k\rangle\|\leqslant \varepsilon\ (1\leqslant k\leqslant n)\}, \]

each of which is determined by a finite set of elements \(y_k\in E'_{(A)}\) \((1\leqslant k\leqslant n)\) and a number \(\varepsilon>0\).

Theorem 2. The module \(L_{\mathfrak B(A)}(E_{(A)},\mathfrak B(A))\) of all \(\mathfrak B(A)\)-linear forms on \(E_{(A)}\), continuous in the weak topology \(\sigma(E_{(A)},E'_{(A)})\), is canonically identifiable with the module \(E'_{(A)}\): every continuous \(\mathfrak B(A)\)-linear form on \(E_{(A)}\) is uniquely representable in the form \(x\mapsto\langle x,y\rangle\), where \(y\in E'_{(A)}\).

Indeed, let \(g\) be an arbitrary continuous \(\mathfrak B(A)\)-linear form on \(E_{(A)}\). Then \(g(bx)=bg(x)\), whatever \(b\in\mathfrak B(A)\) and \(x\in E_{(A)}\); consequently, \(g_\alpha(bx)=b(\alpha)g_\alpha(x)\), where \(g_\alpha=\varepsilon_\alpha\circ g\) \((\alpha\in A)\). In particular, \(g_\alpha(x)=0\) if \(x_\alpha=0\); in other words, for each \(\alpha\in A\) there is defined a mapping \(x_\alpha\mapsto g_\alpha(x)\), which is a linear form on the vector space \(E''\), continuous in the topology \(\sigma(E'',E')\). Therefore \(g_\alpha(x)=\langle x_\alpha,y_\alpha\rangle\), where \(y_\alpha\in E'\) \((\alpha\in A)\). On the other hand, by virtue of the continuity of the form \(g\) in the topology \(\sigma(E_{(A)},E'_{(A)})\) there exist \(z_k\in E'_{(A)}\) \((1\leqslant k\leqslant n)\) such that \(|\langle x_\alpha,y_\alpha\rangle|\leqslant 1\) \((\alpha\in A)\) if \(\|\langle x,z_k\rangle\|\leqslant 1\) \((1\leqslant k\leqslant n)\). Since the family of linear forms \(z_{k\alpha}=\varepsilon_\alpha\circ z_k\) \((1\leqslant k\leqslant n;\ \alpha\in A)\) is equicontinuous on \(E\), for every \(s\) from some neighborhood of zero in \(E\) one has \(|\langle s,z_{k\alpha}\rangle|\leqslant 1\) \((1\leqslant k\leqslant n;\ \alpha\in A)\), whence \(|\langle s,y_\alpha\rangle|\leqslant 1\) \((\alpha\in A)\). Thus the family \((y_\alpha)\) is equicontinuous on \(E\) and, consequently, defines some element \(y\in E'_{(A)}\), for which \(\varepsilon_\alpha\circ y=y_\alpha\) \((\alpha\in A)\) and \(g(x)=\langle x,y\rangle\) \((x\in E_{(A)})\). If \(g(x)=\langle x,z\rangle\) \((x\in E_{(A)})\) for some \(z\) in \(E'_{(A)}\), then, in view of the duality between \(E_{(A)}\) and \(E'_{(A)}\), from \(\langle x,y-z\rangle=0\) \((x\in E_{(A)})\) it follows that \(y=z\), which completes the proof of the theorem.

By analogy with the preceding, endow the module \(E'_{(A)}\) with the weak topology \(\sigma(E'_{(A)},E_{(A)})\), i.e. the weakest of the topologies for which all \(\mathfrak B(A)\)-linear forms of the form \(y\mapsto\langle x,y\rangle\) \((x\in E_{(A)})\) are continuous. Then the following holds.

Theorem 3. The module \(L_{\mathfrak B(A)}(E'_{(A)},\mathfrak B(A))\) of all continuous \(\mathfrak B(A)\)-linear forms on \(E'_{(A)}\) is canonically identifiable with the module \(E_{(A)}\): every continuous \(\mathfrak B(A)\)-linear form on \(E'_{(A)}\) is uniquely representable in the form \(y\mapsto\langle x,y\rangle\), where \(x\in E_{(A)}\).

4. Whatever the nonempty set \(A\), the linear mapping \(i:E\to E_{(A)}\) is continuous if \(E\) is endowed with the original topology, and \(E_{(A)}\) with the weak topology \(\sigma(E_{(A)},E'_{(A)})\).

Theorem 4. Whatever the separated locally convex space \(E\), there exists a set \(A\) such that the mapping \(i\) is a monomorphism.

Indeed, let \(U\) be an arbitrary closed, balanced, convex neighborhood of zero in \(E\); then \(U=M^0\), where \(M\) is some equicontinuous set in \(E'\). We choose the set \(A\) so that its cardinality majorizes the cardinality of any equicontinuous set in \(E'\). In this case \(\operatorname{Card}(M)\leq \operatorname{Card}(A)\), and the set \(M\) can be represented parametrically as an equicontinuous family of linear forms \(y_\alpha\) \((\alpha\in A)\) on \(E\). The family \((y_\alpha)\) determines an element \(y\in E'_{(A)}\) such that \(\varepsilon_\alpha\circ y=y_\alpha\) \((\alpha\in A)\). Let \(s\) be an element in \(E\) such that \(i(s)\in W(y;1)\). Then \(|\langle s,y_\alpha\rangle|\leq 1\) \((\alpha\in A)\), and conversely. In other words,
\[ i(U)=W(y;1)\cap i(E), \]
and, consequently, \(i\) is an isomorphism of the space \(E\) onto \(i(E)\).

The linear mapping \(j:E'\to E\) is continuous if the space \(E'\) is endowed with the strong topology, and \(E'_{(A)}\) with the weak topology \(\sigma(E'_{(A)},E_{(A)})\).

Analogously to Theorem 4, one proves

Theorem 5. Whatever the separated locally convex space \(E\), there exists a set \(A\) such that the mapping \(j\) is a monomorphism.

1. Let \(f:E\to F\) be a continuous linear mapping of the space \(E\) into the separated locally convex space \(F\). Then a \(\mathfrak{B}(A)\)-linear mapping may be defined
   \[ f_{(A)}:E_{(A)}\to F_{(A)}, \]
   called associated with \(f\), and assigning to each continuous linear mapping \(x:E'\to \mathfrak{B}(A)\) the composition \(x\circ f\), which is a continuous linear mapping of the space \(F'\) into \(\mathfrak{B}(A)\).

Analogously, a \(\mathfrak{B}(A)\)-linear mapping may be defined
\[ {}^{t}f_{(A)}:F'_{(A)}\to E'_{(A)}, \]
called associated with the mapping \({}^{t}f:F'\to E'\), conjugate to \(f\), and assigning to each continuous linear mapping \(y:F\to \mathfrak{B}(A)\) the composition \(y\circ f\), which is a continuous linear mapping of the space \(E\) into \(\mathfrak{B}(A)\).

It follows directly from these definitions that the mapping \({}^{t}f_{(A)}\) is conjugate to \(f_{(A)}\) with respect to the canonical \(\mathfrak{B}(A)\)-bilinear forms defined respectively on the products \(F_{(A)}\times F'_{(A)}\) and \(E_{(A)}\times E'_{(A)}\):
\[ \langle f_{(A)}(x),y'\rangle=\langle x,{}^{t}f_{(A)}(y')\rangle, \]
whatever \(x\in E_{(A)}\) and \(y'\in F'_{(A)}\). In particular, the mappings \(f_{(A)}\) and \({}^{t}f_{(A)}\) are continuous in the weak topologies.

Theorem 6. Let \(f:E\to F\) be a continuous linear mapping. In order that the continuous \(\mathfrak{B}(A)\)-linear mapping \(\xi:E_{(A)}\to F_{(A)}\) enter into the commutative diagram
\[ \begin{array}{ccc} E & \xrightarrow{\,f\,} & F\\ {\scriptstyle i}\downarrow & & \downarrow{\scriptstyle i}\\ E_{(A)} & \xrightarrow{\,\xi\,} & F_{(A)} \end{array} \]
it is necessary and sufficient that \(\xi\) coincide with \(f_{(A)}\).

Indeed, the diagram holds for the mapping \(f_{(A)}\) associated with \(f\); if it also holds for some continuous \(\mathfrak{B}(A)\)-linear mapping \(\xi\), then \((f_{(A)}-\xi)\circ i=0\). Therefore
\[ g\circ (f_{(A)}-\xi)=0 \]
for any \(\mathfrak{B}(A)\)-linear form \(g\) on \(F_{(A)}\), continuous in the weak topology \(\sigma(F_{(A)},F'_{(A)})\); consequently, \(f_{(A)}-\xi=0\).

Theorem 7. In order that the continuous \(\mathcal B(A)\)-linear mapping
\(\xi:\ F'_{(A)}\to E'_{(A)}\) enter into the commutative diagram

\[ \begin{array}{ccc} F' & \xrightarrow{\ t_f\ } & E'\\ j\downarrow & & \downarrow j\\ F'_{(A)} & \xrightarrow{\ \xi\ } & E'_{(A)} \end{array} \]

it is necessary and sufficient that \(\xi\) coincide with \({}^{t}f_{(A)}\).

1. Let \(\sigma(E,E'_{(A)})\) be the weakest of the topologies in \(E\) for which all linear mappings \(E\to \mathcal B(A)\) belonging to \(E'_{(A)}\) are continuous. If \(A\subset A'\), then the topology \(\sigma(E,E'_{(A')})\) is weaker than \(\sigma(E,E'_{(A)})\). It can be shown that every linear mapping \(f:E\to F\), continuous for the topologies \(\sigma(E,E'_{(A')})\), \(\sigma(F,F'_{(A')})\), is also continuous for the topologies \(\sigma(E,E'_{(A)})\), \(\sigma(F,F'_{(A)})\). In particular, if \(f\) is continuous for the original topologies in \(E\) and \(F\), then it is continuous also for the topologies \(\sigma(E,E'_{(A)})\), \(\sigma(F,F'_{(A)})\), whatever the nonempty set \(A\) may be.

Theorem 8. Let \(f:E\to F\) be a linear mapping, continuous for the topologies \(\sigma(E,E'_{(A)})\), \(\sigma(F,F'_{(A)})\). In order that \(f\) be a monomorphism for the same topologies, it is necessary and sufficient that the mapping
\[ {}^{t}f_{(A)}:\ F'_{(A)}\to E'_{(A)}, \]
associated with \({}^{t}f\), be surjective.

Indeed, if \(f\) is a monomorphism, then, by the Hahn–Banach theorem, every continuous linear mapping \(x:E\to \mathcal B(A)\) can be represented in the form \(x=y\circ f\), where \(y\in F'_{(A)}\); hence the mapping \({}^{t}f_{(A)}\) is surjective. Conversely, if \({}^{t}f_{(A)}\) is surjective, then to every \(x\in E'_{(A)}\) there corresponds a \(y\in F'_{(A)}\) such that \(x=y\circ f\); consequently, \(f\) is injective. Since, moreover,
\(\langle s,\varepsilon_\alpha\circ x\rangle=\langle f(s),\varepsilon_\alpha\circ y\rangle\)
\((s\in E;\ \alpha\in A)\), it follows that \(f\) is a monomorphism.

Theorem 9. Let \(A\subset A'\). If the mapping \(f:E\to F\) is a monomorphism for the topologies \(\sigma(E,E'_{(A')})\), \(\sigma(F,F'_{(A')})\), then \(f\) is a monomorphism also for the topologies \(\sigma(E,E'_{(A)})\), \(\sigma(F,F'_{(A)})\).

Indeed, \(f\) is continuous for the topologies \(\sigma(E,E'_{(A)})\), \(\sigma(F,F'_{(A)})\). Every continuous linear mapping \(x:E\to \mathcal B(A)\) can be represented in the form \(x=\theta\circ x'\), where
\(\theta:\mathcal B(A')\to \mathcal B(A)\) is the restriction mapping, and
\(x':E\to \mathcal B(A')\) is some continuous linear mapping. Since \(f\) is a monomorphism for the topologies \(\sigma(E,E'_{(A')})\), \(\sigma(F,F'_{(A')})\), we have \(x'=y'\circ f\), where \(y'\in F'_{(A')}\). Then \(x=y\circ f\), where \(y=\theta\circ y'\) is a continuous linear mapping \(F\to \mathcal B(A)\).

Corollary 1. If the mapping \(f:E\to F\) is a monomorphism for the original topologies, then it is a monomorphism also for the topologies
\(\sigma(E,E'_{(A)})\), \(\sigma(F,F'_{(A)})\), whatever the nonempty set \(A\) may be.

Indeed, by Theorem 4, for some \(A'\supset A\) the topologies
\(\sigma(E,E'_{(A')})\), \(\sigma(F,F'_{(A')})\) coincide with the original topologies in \(E\) and \(F\), respectively.

Corollary 2. In order that a continuous linear mapping
\(f:E\to F\) be a monomorphism (for the original topologies), it is necessary and sufficient that, for every nonempty set \(A\), the mapping
\[ {}^{t}f_{(A)}:\ F'_{(A)}\to E'_{(A)} \]
be surjective.

Kharkov State University
named after A. M. Gorky

Received
16 VI 1966