UDC 513.83 + 513.88 + 519.50

MATHEMATICS

G. Kh. BERMAN

EXTENSION OF FUNCTORS FROM THE CATEGORY OF BANACH SPACES TO THE CATEGORY OF \((\alpha)\)-SPACES

(Presented by Academician P. S. Novikov on 11 VI 1966)

1°. Let \(E\) be a separable locally convex space and \(A\) a bounded absolutely convex set (a ball) in \(E\). By \(E^{A}\) we shall denote the linear hull of \(A\), endowed with the norm
\[ \|x\|_{A}=\inf\{\rho>0:\ x\in \rho A\}. \]
The canonical embedding \(i^{A}: E^{A}\to E\) is continuous.

If \(E^{A}\) is a Banach space, then \(A\) is called a Banach ball \((^{1})\), and \(E^{A}\) a Banach subspace in \(E\). The space \(E\) is called locally complete \((^{1})\) if every sequence of its elements contained in some \(E^{A}\) and forming there a Cauchy sequence converges in \(E\). A separable locally convex space \(E\) is called a space of type \((\alpha)\), or an \((\alpha)\)-space \((^{1})\), if it is locally complete and is a \((\beta)\)-space \((^{4})\), i.e. representable as an inductive limit of Banach spaces.

The class of all \((\alpha)\)-spaces together with all continuous linear mappings of each of them into each forms a category, which we shall denote by \(\mathfrak A\). The class \(\mathfrak B\) of all Banach spaces constitutes a full subcategory in \(\mathfrak A\).

In the present note we prove the possibility of extending any functor in the category \(\mathfrak B\) (or defined on \(\mathfrak B\) and taking values in \(\mathfrak A\)) to a functor in the category \(\mathfrak A\). Such an extension is, generally speaking, non-unique. The existence of two “extreme” extensions is established. As a consequence, in particular, one obtains a certain description of the totality of functors in \(\mathfrak A\) whose restrictions to \(\mathfrak B\) coincide.

2°. Spaces of type \((\alpha)\) were first singled out and studied by D. A. Raikov in \((^{1})\). We recall some facts and consequences from that work.

Proposition 1. Let \(\{A\}\) be a complete system, i.e. the system of all Banach balls of the \((\alpha)\)-space \(E\). Then \(E\) is the limit of the direct spectrum of its Banach subspaces \(\{E^{A}\}\) with respect to the embeddings
\[ i^{A}:E^{A}\to E. \]

Every separable locally convex space can be locally completed \((^{1})\). For spaces of countable character, in particular for normed spaces, local completion coincides with ordinary completion.

The topology of any separable locally convex space \(E\) can be \((\beta)\)-strengthened, i.e. can be strengthened so that, in the new topology, \(E\) becomes a \((\beta)\)-space with the same stock of Banach balls.

Local completeness is stable with respect to \((\beta)\)-strengthening of the topology. A locally completed \((\alpha)\)-space is again an \((\alpha)\)-space.

Let \(E\) and \(F\) be locally convex spaces. By \(L(E,F)\) we shall denote the vector space of all continuous linear mappings of \(E\) into \(F\). The space \(L(E,F)\) in the topology of simple convergence will be denoted by \(L_{s}(E,F)\), and its \((\beta)\)-strengthening by \(L_{i}(E,F)\). The topology of the space \(L_{i}(E,F)\) is called inductive \((^{1})\). From Theorems 4.4 and 4.8 of \((^{1})\) it follows immediately ...

Proposition 2. If \(E\) is a space of type \((\alpha)\), and \(F\) is locally complete, then \(L_i(E,F)\) is an \((\alpha)\)-space.

In the category of separable locally convex spaces an inverse spectrum always has a limit. By the limit of a direct spectrum in this same category one understands the separable space associated with the limit of this same spectrum in the category of all locally convex spaces.

Proposition 3. The limit of a direct (respectively, inverse) spectrum in the category \(\mathfrak A\) is the locally completed (respectively, \((\beta)\)-strengthened) limit of the same spectrum in the category of separable locally convex spaces.

Let \(\{A\}\) and \(\{B\}\) be complete systems of Banach disks in \((\alpha)\)-spaces \(E\) and \(F\), respectively. The tensor products \(E^A \widehat{\otimes} F^B\) \((^4)\) form a direct spectrum of Banach spaces. The limit of this direct spectrum in the category \(\mathfrak A\) is called the tensor \((\alpha)\)-product of the spaces \(E\) and \(F\) \((^1)\) and is denoted by \(E \widehat{\otimes} F\). The space \(E \widehat{\otimes} F\), obviously, is an \((\alpha)\)-space.

\(3^0\). In the definition of a (covariant) functor \(S:\mathfrak C\to\mathfrak A\) (\(\mathfrak C\) is a subcategory of \(\mathfrak A\)), in addition to the known conditions, we shall assume that the mapping \(\varphi\to S\varphi\) from the space \(L_i(E,F)\) into the space \(L_i(SE,SF)\), generated by the functor \(S\), is linear and continuous.

Basic functors. Let \(E\) be a fixed space of type \((\alpha)\). The correspondence \(F\to L_i(E,F)\), where \(F\in\mathfrak A\), defines a functor (Proposition 2) in the category \(\mathfrak A\), which will be denoted by \(\Omega_E\). On morphisms \(\Omega_E\) is defined in the natural way. The tensor \((\alpha)\)-product defines a functor \(\Sigma_E\) as follows: \(\Sigma_EF=E\widehat{\otimes}F\).

It was established in \((^1)\) that the category \(\mathfrak A\) with the functors \(\Sigma_E\) and \(\Omega_E\) forms a \(D\)-category in the sense of A. S. Schwarz \((^2)\). Since the functor \(\Sigma_E\) is left adjoint to the functor \(\Omega_E\) (and \(\Omega_F\) to \(\Sigma_E\) is right) in the sense of Kan \((^6)\), it follows that

Proposition 4 \((^2)\). The functor \(\Sigma_E\) (respectively, \(\Omega_E\)) is permutable with the limits of direct (respectively, inverse) spectra in the category \(\mathfrak A\).

\(4^0\). Consider a functor \(T:\mathfrak B\to\mathfrak B\) (or \(T:\mathfrak B\to\mathfrak A\)). The collection \(P_T\) of all its extensions to a functor in \(\mathfrak A\) forms a category if, as \(\operatorname{Hom}(T^\gamma,T^\delta)\) \((T^\gamma,T^\delta\in P_T)\), one takes all those mappings of \(T^\gamma\) into \(T^\delta\) whose restrictions to \(\mathfrak B\) coincide with the identity mapping of the functor \(T\) into itself.

Main theorem. Every functor \(T:\mathfrak B\to\mathfrak B\) (or \(T:\mathfrak B\to\mathfrak A\)) admits an extension to a functor in \(\mathfrak A\), and in the category \(P_T\) of all its extensions there is an initial object \(T^i\) and a final object \(T^\pi\), i.e. for any \(T^\gamma\in P_T\) there exists a unique mapping \(\sigma^{i,\gamma}:T^i\to T^\gamma\) and a unique mapping \(\sigma_{\gamma,\pi}:T^\gamma\to T^\pi\).

The functor \(T^i\) is constructed as follows. Let \(E\) be an \((\alpha)\)-space and let \(\{E^A\}\) be the direct spectrum of its Banach subspaces determined by a complete system of Banach disks \(\{A\}\) in \(E\) (Proposition 1). By \(T^iE\) we denote the limit of the corresponding direct spectrum \(\{TE^A\}\) in \(\mathfrak A\) (Proposition 3).

Let \(E\) be a separable locally convex space and let \(\{U\}\) be a fundamental system of its closed absolutely convex neighborhoods of zero. By \(E_U\) we shall mean the quotient space \(E/N_U\) (\(N_U\) is the largest subspace contained in \(U\)), endowed with the usual normed topology. Denote the completion of \(E_U\) by \(\widehat E_U\). The system of Banach spaces \(\{\widehat E_U\}\) forms an inverse spectrum, whose limit in the category of locally convex spaces coincides with the completion \(\widehat E\) of the space \(E\) \((^5)\). If \(E\) is a space of type \((\alpha)\), then the space \(T^\pi E\) is defined as the limit of the inverse spectrum \(\{T\widehat E_U\}\) in the category \(\mathfrak A\) (Proposition 3).

Denote by \(F(\mathfrak A)\) the category of all functors acting in \(\mathfrak A\). Let \(S \in F(\mathfrak A)\), and let \(S|_{\mathfrak B}\) be the restriction of \(S\) to \(\mathfrak B\). From the main theorem it follows immediately:

Proposition 5. The category \(P_{S|_{\mathfrak B}}\) forms a spectrum \({}^{(2)}\) in \(F(\mathfrak A)\), whose projective limit is the initial object \(S^i\), and whose inductive limit is the final object \(S^\pi\).

5°. Proposition 6. If the functor \(S:\mathfrak A \to \mathfrak A\) is permutable with limits of direct spectra in \(\mathfrak A\), then \(S^i = S\).

Proposition 7. If the functor \(S:\mathfrak A \to \mathfrak A\) is permutable with limits of inverse spectra in \(\mathfrak A\), then \(S\) and \(S^\pi\) coincide on complete spaces of the category \(\mathfrak A\).

Example 1. Denote by \(I\) the identity functor in the category \(\mathfrak A\). From Proposition 6 it follows that \(I^i = I\). If \(E\) is a complete \((\alpha)\)-space, then (Proposition 7) \(I^\pi E = E\). In the general case, to obtain \(I^\pi E\) one must complete \(E\) and then \((\beta)\)-strengthen its topology.

Example 2. From Propositions 5 and 6 it follows that \(\Sigma_E^i = \Sigma_E\).

Example 3. For the functor \(\Omega_E^\pi\) the equality
\[ \Omega_E^\pi(F)=L_i(E,\hat F) \]
holds, where \(\hat F\) is the completion of the \((\alpha)\)-space \(F\). In particular, if \(F\) is complete, then (Propositions 5 and 7) \(\Omega_E^\pi(F)=L_i(E,F)\). The set of all mappings \(\varphi \in L(E,F)\) that transform some neighborhood of zero in \(E\) into a bounded set in \(F\) coincides with the union of the spaces \(L_i(E,F^B)\) (\(\{B\}\) is a complete system of Banach disks in \(F\)). The limit of the direct spectrum \(\{L_i(E,F^B)\}\) in the category \(\mathfrak A\) coincides with the space \(\Omega_E^i(F)\). In particular, if \(E\) is a Banach space, then \(\Omega_E^i=\Omega_E\).

Example 4. Each space of numerical sequences \(l_p\) \((1 \le p \le \infty)\) defines in the category \(\mathfrak B\) the corresponding functor \(\bar l_p\) \({}^{(3)}\). Let \(E\) be a space of type \((\alpha)\). Consider the space of all sequences \((x_n)\) from the completion \(\hat E\) for which the series
\[ \sum_{n=1}^{\infty}\alpha_n x_n \]
converges absolutely in \(\hat E\), whatever numerical sequence \((\alpha_n)\in l_q\), \(p^{-1}+q^{-1}=1\). As a set of elements this space coincides with \(\bar l_p^\pi(E)\). The set of all sequences \((x_n)\) from \(E\) for which the series
\[ \sum_{n=1}^{\infty}\alpha_n x_n \]
converges absolutely in some fixed Banach subspace of \(E\), whatever numerical sequence \((\alpha_n)\in l_q\), \(p^{-1}+q^{-1}=1\), is naturally turned into a Banach space. The limit of the direct spectrum in the category \(\mathfrak A\) of Banach spaces obtained in this way coincides with \(\bar l_p^i(E)\).

In conclusion, the author expresses his deep gratitude to Prof. D. A. Raikov for his attention to the work, very valuable discussions, and comments.

Beltsy State Pedagogical Institute
named after Alecu Russo

Received
26 V 1966

REFERENCES

\({}^{1}\) D. A. Raikov, Mat. sbornik, 67 (109), 2 (1965).
\({}^{2}\) A. A. Shvarts, DAN, 148, No. 2 (1963).
\({}^{3}\) B. S. Mityagin, A. S. Shvarts, UMN, 19, 1 (116) (1964).
\({}^{4}\) A. Grothendieck, Mem. Am. Math. Soc., No. 16 (1955).
\({}^{5}\) G. Köthe, Topologische lineare Räume, Berlin, 1960.
\({}^{6}\) D. M. Kan, Sborn. per. Matematika, 3, 2 (1959).