UDC 513.88:513.83

MATHEMATICS

V. A. GEILER

ON FUNCTORS DEFINED BY REFLEXIVE \(K\)-SPACES

(Presented by Academician L. V. Kantorovich on 24 I 1969)

Functors on the category of locally convex spaces were studied by G. H. Berman \((^{1,2})\). He constructed a duality theory for functors on the category \(\mathcal{F}\mathcal{N}\) of nuclear Fréchet spaces. In particular, he studied the functor \(\Lambda_{\mathfrak M}\), defined by the perfect space of sequences \(\lambda\), which, from the point of view of the theory of \(K\)-spaces, is a discrete reflexive \(K\)-space of countable type \((^{3,4})\). In the present note, for each reflexive \(K\)-space \(X\), endowed with a certain locally convex topology, a functor \(\Phi_X^{\mathfrak M}\) is constructed, of which \(\Lambda_{\mathfrak M}\) is a special case. We shall also consider a certain functor \(\Omega_A^{\mathfrak S}\), and it turns out that on certain categories (containing \(\mathcal{F}\mathcal{N}\)) functors of this type are isomorphic to \(\Omega_A\), \(\Sigma_A\), \(\Phi_X^{\mathfrak M}\), and also to the \(\varepsilon_A\) introduced in \((^5)\).

In what follows \(X\) denotes a reflexive \(K\)-space, \(\bar X\) its conjugate, \(\mathfrak M\) a certain system of bounded absolutely convex normal subsets of \(\bar X\) covering \(\bar X\) (and called a topologizing system for \(X\)); if all the sets in \(\mathfrak M\) are \(\sigma(\bar X,X)\)-bicompact, then \(\mathfrak M\) is called bicompact. By \(X_{\mathfrak M}\) we denote the space \(X\), endowed with the locally convex topology generated by the polars of the sets in \(\mathfrak M\). Using the results of \((^6)\), it is not difficult to show that among the spaces obtained in this way there is \(X_\tau\)—the space \(X\) endowed with the Mackey topology \(\tau(X,\bar X)\); we also note that a subset of \(X_{\mathfrak M}\) is bounded if and only if it belongs to the system \(\mathfrak B\) of \(\sigma(X,\bar X)\)-bounded subsets of \(X\). Using the results of A. G. Pinsker \(((^4),\) Chap. IV, Theorems 1.31, 4.13), we realize \(X\) as the base of a \(K\)-space \(L^1_{\mathrm{loc}}(T;\mu)\), where \(T\) is a locally bicompact space which is a sum of extremally disconnected bicompacts; \(\mu\) is a positive measure on \(T\) such that the families of locally \(\mu\)-negligible \((^7)\) and nowhere dense subsets of \(T\) coincide, and \(L^1_{\mathrm{loc}}(T;\mu)\) is the quotient space of the space \(\mathscr L^1_{\mathrm{loc}}(T;\mu)\) of locally summable functions by the subspace of locally \(\mu\)-negligible functions. By \(\theta\) we shall denote the canonical mapping of \(\mathscr L^1_{\mathrm{loc}}(T;\mu)\) onto \(L^1_{\mathrm{loc}}(T;\mu)\).

Let \(E\) denote a separable locally convex space, and \(\mathfrak U\) the family of its absolutely convex neighborhoods of zero. By \(\mathscr X_{\mathfrak M}(E)\) (respectively \(\mathscr X_{\mathfrak M}[E]\)) we denote the space of \(\mu\)-measurable functions from \(T\) into \(E\) such that for every continuous seminorm \(p\) on \(E\), \(\theta(pf)\in X\) (for every \(a'\in E'\), \(\theta(a'f)\in X\)), endowed with the topology generated by the seminorms

\[ f\mapsto p_{M^0}(\theta(p_U f)) \]

(respectively

\[ f\mapsto \sup_{a'\in U^0} p_{M^0}(\theta(a'f)), \]

where \(M\in\mathfrak M\), \(U\in\mathfrak U\). These spaces do not depend on the choice of the realization of \(X\). By \(X_{\mathfrak M}(E)\) (respectively \(X_{\mathfrak M}[E]\)) we denote the separable locally convex space associated with \(\mathscr X_{\mathfrak M}(E)\) \((\mathscr X_{\mathfrak M}[E])\).

Theorem 1. If \(E\) is a nuclear space, then \(X_{\mathfrak M}(E)\) is isomorphic to \(X_{\mathfrak M}[E]\).

If \(E,F\) are locally convex spaces and \(\mathfrak S\) is a system of bounded subsets of \(E\), then \(L^{\mathfrak S}(E,F)\) is the space of continuous li-

linear maps from \(E\) to \(F\), endowed with the \(\mathfrak S\)-topology; by \(\mathfrak B\) we shall denote the system of all bounded subsets of \(E\), and by \(\mathfrak E\) that of all equicontinuous subsets of \(E'\).

Theorem 2. Let \(E\) be a complete nuclear space, and let \(\mathfrak M\) be a bicompact topologizing system for \(\bar X\).

Each of the following conditions is sufficient for \(X_{\mathfrak M}(E)\) to be isomorphic to \(L_{\mathfrak M}(X_{\mathfrak M}, E)\): a) \(E\) is dually nuclear \((^8)\); b) let \(\mathfrak a\) be the character of \(E\); if \(\mathfrak F\) is a collection of nowhere dense subsets \(T\) and \(\operatorname{card}(\mathfrak F)\leq \mathfrak a\), then \(\bigcup \mathfrak F\) is nowhere dense.

Corollary. If, under the hypotheses of Theorem 2, the system \(\mathfrak M\) is bicompact, then
\[ X_{\mathfrak M}(E)\simeq X_{\mathfrak M}\hat\otimes E. \]

In what follows, \(\mathscr L\) denotes the category of separated locally convex spaces, \(\mathscr E\mathscr L\) the full subcategory of complete spaces in \(\mathscr L\), and \(\mathscr I\mathscr T\) the category of infrabarreled spaces; we shall also consider the following full subcategories of \(\mathscr E\mathscr L\): \(\mathscr B\mathscr L\mathscr T\), of barrelled spaces; \(\mathscr N\), of nuclear spaces; \(\mathscr D\mathscr N\), of nuclear dually nuclear spaces; \(\mathscr R\mathscr N\), of reflexive nuclear spaces; and \(\mathscr R\mathscr D\mathscr N\), of reflexive spaces in \(\mathscr D\mathscr N\).

For each space \(X_{\mathfrak M}\) we define a functor \(\Phi_X^{\mathfrak M}\) on \(\mathscr L\) as follows: \(\Phi_X^{\mathfrak M}E=X_{\mathfrak M}(E)\), and for \(\alpha\in L(E,F)\), \(\Phi_X^{\mathfrak M}\alpha\) is the mapping induced by passage to quotient spaces from the mapping \(f\mapsto \alpha f\) from \(\mathscr X_{\mathfrak M}(E)\) to \(\mathscr X_{\mathfrak M}(F)\). The functors \(\Omega_A\) and \(\Sigma_A\) are the same as in (1). Let \(A\in\mathscr L\); \(\mathfrak S\) is a system of bounded subsets of \(A\) covering \(A\); define the functor \(\Omega_A^{\mathfrak S}\) on the category \(\mathscr K\) by
\[ \Omega_A^{\mathfrak S}E=L_{\mathfrak S}(A,E),\qquad \Omega_A^{\mathfrak S}\alpha=\Omega_A\alpha\quad (\alpha\in L(E,F)). \]
The functor \(\Omega_{A_c}^{\mathfrak E}\) will also be denoted by \(\Psi_A\).

The following assertions are valid:

a) if \(A\in\mathscr B\mathscr L\), then \(\Sigma_A\) is isomorphic to \(\Psi_A\) on \(\mathscr N\); if \(A\in\mathscr N\), then \(\Sigma_A\) is isomorphic to \(\Psi_A\) on \(\mathscr B\mathscr L\);

b) for every \(A\in\mathscr L\), \(\varepsilon_A\) is isomorphic to \(\Omega_{A_c}^{\mathfrak E}\) on \(\mathscr L\);

c) on \(\mathscr D\mathscr N\), \(\Phi_X^{\mathfrak M}\) is isomorphic to \(\Omega_{\bar X_\tau}^{\mathfrak M}\).

Proof. Assertion a) follows from Grothendieck’s theorem \((^9,\ \text{Ch. II, Theorem 6})\); b) from Corollary 2 to Proposition 4 \((^{10})\); c) from Theorem 2.

If \(R\) and \(S\) are such functors that the class of mappings from \(R\) to \(S\) is a set, then by \(H_b(R,S)\) we shall denote the space of mappings from \(R\) to \(S\), endowed with the initial topology with respect to the family of mappings
\[ P_E:H(R,S)\to L_{\mathfrak B}(RE,SE) \tag{1}. \]
A functor \(R\) is called continuous on the category \(\mathscr K\) if, for every pair of spaces \(E,F\) from \(\mathscr K\), the mapping
\[ R:L_{\mathfrak B}(E,F)\to L_{\mathfrak B}(RE,RF) \]
is continuous; \(R\) is called regular on \(\mathscr K\) if \(H_b(\Omega_A,R)\) is canonically isomorphic to the space \(RA\) for every \(A\) in \(\mathscr K\) \((^1)\).

Theorem 3. For every \(A\in\mathscr L\), the functor \(\Omega_A^{\mathfrak S}\) is continuous on \(\mathscr L\) and regular on \(F\mathscr T\).

Theorem 4. Let \(A,B\in\mathscr L\); let \(\mathfrak S\) (respectively \(\mathfrak T\)) be a system of absolutely convex weakly bicompact subsets of \(A\) (respectively of \(B\)).

If for every \(u\in L(A_{\mathfrak S}',B_{\mathfrak T}')\) the adjoint mapping \(u'\in L(B,A)\) (in particular, if \(B\) is a Mackey space), then
\[ H_b(\Omega_A^{\mathfrak S},\Omega_B^{\mathfrak T}) \]
is isomorphic to
\[ L_{\mathfrak B}(A_{\mathfrak S}',B_{\mathfrak T}'). \]

By \(\mathscr D S\) is denoted the functor dual \((^1)\) to \(S\).

Theorem 5. Let \(A\in\mathscr B\mathscr L\), and let the sets in \(\mathfrak S\) be relatively weakly bicompact. On the category \(\mathscr N\) we have: a) \(\mathscr D\Omega_A^{\mathfrak S}=\Omega_{A_{\mathfrak S}'}\); b) if \(A_\tau\) is barrelled, then
\[ \mathscr D^2\Omega_A^{\mathfrak S}=\Omega_{A_\tau}; \]
c) if \(A\) is reflexive, then \(\Omega_A\) is reflexive.

Let us consider some consequences of these theorems.

Corollary 1. *Let \(X,Y\) be reflexive \(K\)-spaces, and let \(\mathfrak M\) (co-

respectively \(\mathfrak N\)) is a topologizing system for \(X\) (respectively \(Y\)); the functors \(\Phi_X^{\mathfrak M}\) and \(\Phi_Y^{\mathfrak N}\) are defined on \(\mathcal D\mathcal N\).

Then: a) \(\Phi_X^{\mathfrak M}\) is continuous on \(\mathcal D\mathcal N\) and regular on \(\mathcal F\mathcal D\mathcal M\); b) if the systems \(\mathfrak M\) and \(\mathfrak N\) are bicompact, then \(H_b(\Phi_X^{\mathfrak M}, \Phi_Y^{\mathfrak N})\) is isomorphic to \(L_{\mathfrak B}(X_{\mathfrak M}, Y_{\mathfrak N})\); c) if \(\mathfrak M\) is bicompact, then \(\mathcal D\Phi_X^{\mathfrak M}=\Phi_X^{\mathfrak B}\); if \(X_{\mathfrak M}\) is semireflexive, then \(\mathcal D^2\Phi_X^{\mathfrak M}=\Phi_X^{\mathfrak B}\); if \(X_{\mathfrak M}\) is a reflexive locally convex space, then \(\Phi_X^{\mathfrak M}\) is reflexive.

Corollary 2. Let \(A,B\in \mathcal C\mathcal L\) (respectively \(\mathcal N\)); the functors \(\Sigma_A,\Sigma_B,\Omega_A\) are defined on \(\mathcal N\) (respectively \(\mathcal C\mathcal L\)).

Then: a) \(\Sigma_A\) is continuous on \(\mathcal N\) (respectively \(\mathcal C\mathcal L\)) and regular on \(\mathcal R\mathcal N\) (respectively \(\mathcal L\mathcal T\)); \(\Omega_A\) is regular on \(\mathcal L\mathcal T\); b) \(H_b(\Sigma_A,\Sigma_B)\) is isomorphic to \(L_{\mathfrak B}(A,B)\); c) \(\mathcal D\Omega_A=\Sigma_A\) on \(\mathcal N\) (respectively \(\mathcal C\mathcal L\)); \(\mathcal D\Sigma_A=\Omega_A\) on \(\mathcal R\mathcal N\) (respectively \(\mathcal L\mathcal T\)); d) \(\Sigma_A\) and \(\Omega_A\) are reflexive on \(\mathcal R\mathcal N\) (respectively \(\mathcal L\mathcal T\)); if \(A\) is reflexive, then \(\Sigma_A\) is reflexive on \(\mathcal N\).

Using Theorem 2.5 of paper \((^2)\), we obtain

Corollary 3. For every functor defined on \(\mathcal N\), there exists a continuous dual functor defined on the category of complete spaces satisfying the approximation condition.

Remark 1. Along with the definition of a dual functor given in (1), it is natural also to consider the following definition. Let \(S\) be a functor from \(\mathcal K\) to \(\mathcal L\); by \(\partial S\) denote the functor acting as follows: for \(E\in\mathcal L\),
\[ \partial SE=H_b(S,\Psi_E); \]
if \(u\in L(E,F)\), then
\[ ((\partial Su)\alpha)_X=(\Psi_Xu)\alpha_X, \]
where \(\alpha\in H_b(S,\Psi_E)\),
\[ \alpha_X\in L_{\mathfrak B}(SX,L_{\varepsilon}(E'_c,X))\simeq L_{\mathfrak B}(SX,L_{\varepsilon}(X'_c,E')), \]
\[ (\Psi_Xu)\alpha_X\in L_{\mathfrak B}(SX,L_{\varepsilon}(X'_c,F')\simeq L_{\mathfrak B}(SX,L_{\varepsilon}(F'_c,X)))=L_{\mathfrak B}(SX,\Psi_FX). \]

With such a definition of duality, the notion of reflexivity of functors naturally arises, and it turns out that the functors \(\Omega_A\) and \(\Psi_A\) are dual to one another and are reflexive on the category \(\mathcal L\mathcal T\) in this sense.

Remark 2. We indicate one more generalization of the results of \((^{1,2,5})\). We shall call a functor \(t\) from a category \(\mathcal K\subset\mathcal L\) to the category of partially ordered sets \(O\) topologizing if \(tE\), for every \(E\) in \(\mathcal K\), is a saturated system of absolutely convex bounded subsets of \(E\), and if for \(\alpha\in L(E,F)\) there is a canonical distribution \(\alpha\) on \(tE\). Examples of topologizing functors on \(\mathcal L\): \(b\), \(bE\) is the collection of all bounded subsets of \(E\); \(p\), \(pE\) is the collection of all precompact subsets of \(E\); \(c\), \(cE\) is the saturated system generated by the absolutely convex bicompact subsets of \(E\); \(s\), \(sE\) is the collection of finite-dimensional absolutely convex subsets of \(E\). The pair \(\langle\mathcal K,t\rangle\) will be called a topologized category. A functor \(S:\mathcal K_1\to\mathcal K_2\) will be called continuous from \(\langle\mathcal K_1,t\rangle\) to \(\langle\mathcal K_2,u\rangle\) if all the induced mappings \(L_{tE}(E,F)\to L_{uSE}(SE,SF)\) are continuous. To functors in topologized categories one can extend the notions of regularity, duality, etc.

In conclusion I consider it my pleasant duty to express my deep gratitude to A. G. Pinsker and D. A. Raikov for their great attention to this work.

Mordovian State University

Received
18 I 1969

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