Journal of Function Spaces

Volume 2016, Article ID 3763649, 8 pages

http://dx.doi.org/10.1155/2016/3763649

## Topological Dual Systems for Spaces of Vector Measure -Integrable Functions

^{1}Departamento de Análisis Matemático, Universidad de Valencia, Burjassot, 46100 Valencia, Spain^{2}Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain

Received 21 April 2016; Accepted 30 May 2016

Academic Editor: Miguel Martín

Copyright © 2016 P. Rueda and E. A. Sánchez Pérez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We show a Dvoretzky-Rogers type theorem for the adapted version of the -summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guarantee that the space has to be finite dimensional, contrary to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals, our vector valued version of convergence in the weak topology, is equivalent to the convergence with respect to the norm. Examples and applications are also given.

#### 1. Introduction

Summability in Banach spaces is one of the main topics in Functional Analysis, and results concerning the behavior of summable sequences are fundamental tool for its applications. Comparison between norm and weak absolutely summable series is at the origin of some classical problems in the theory of Banach spaces, and it was the starting point of the theory of -summing operators. In this paper we are interested in providing new elements for the analysis of summability in the case of Banach function spaces by using a vector valued duality that is provided by the vector measure integration theory on spaces of integrable functions with respect to a vector measure . These spaces represent, in fact, all order continuous -convex Banach lattices with weak unit. This theory supplies a distinguished element, the vector valued integral, for the study of summability in Banach spaces of measurable functions. It is well known that whenever and , . In this case, the integral determines a vector valued bilinear map that yields to duality: the vector valued duality between and (see [1, 2]).

This vector valued duality is the framework to study natural topologies on spaces of integrable functions with respect to a vector measure, as the topology generated by the seminorms and , when varying . This new vector valued point of view was first taken into consideration in the study of convergence of sequences: the relation between the convergence of sequences in spaces of vector measure integrable functions and the convergence of the corresponding vector valued integrals has been treated since the seventies (see, e.g., [3, 4], [5, Section ], [6], and the references therein). In this paper we are interested in the summability of sequences in spaces induced by the vector valued duality, that is, when the role played by the weak topology is assumed by the topology . It is worth mentioning that the -convexification () of the space of a vector measure was introduced as a tool for analyzing summability (see [1]), trying to bring together vector valued integration and the theory of -summing operators in Banach spaces (see also [7, 8]).

The classical Dvoretzky-Rogers theorem can be stated as follows: the identity map in a Banach space is absolutely -summing for some , if and only if is finite dimensional. This paper is devoted to proving an extension for Banach function spaces of this result. In our context, the usual scalar duality is replaced by the vector valued duality given by a vector measure and the role of the weak topology in the Banach space is assumed by the topology . In order to develop our study, we analyze some properties of the -summing operators that map summable sequences to norm summable sequences. Our main result shows the necessity of adding some topological requirements on local compactness to characterize finite dimensional spaces in terms of the -summability of the identity map. The last section shows an application to the study of subspaces of that are fixed by the integration operator. As a consequence of our Dvoretzky-Rogers type theorem, we prove that, under the local compactness hypotheses, only finite dimensional subspaces can be fixed by the integration map.

#### 2. Preliminaries

We use standard Banach space notation. Let . Then we write for the extended real number satisfying . We follow the definition of Banach function space over a finite measure given in [9, Def. 1.b.17, p. 28]. Throughout the paper will denote an infinite dimensional Banach function space over ; that is, is a Banach lattice of , a.e., equal classes of -integrable functions with a lattice norm and the a.e. order satisfying . We will also assume that is order continuous; that is, for each decreasing sequence in ,

Let be a real Banach space and let be a measurable space. If is a countably additive vector measure, we write for its range. The variation of is given by , where the supremum is computed over all finite measurable partitions of . is the semivariation of ; that is, , , where is the scalar measure given by . The Rybakov Theorem (see [10, Ch. IX]) establishes that there exists such that is absolutely continuous with respect to a so-called Rybakov measure that means that whenever . For , a (real) measurable function is said to be -integrable with respect to if is integrable with respect to all measures and for each there exists an element such that , .

The space , , is defined to be the Banach lattice of all (-equivalence classes of) measurable real functions defined on that are -integrable with respect to when the a.e. order and the norm are considered. It is an order continuous -convex Banach function space over any Rybakov measure for (see [1, Proposition ]; see also [11] and [6, Ch. ] for more information on these spaces). For the case , is defined as . A relevant fact is that, for each , (see [6, Prop. ] and [1, Sec. ]; see also [11]). Moreover, for each These relations allow defining the so-called vector measure duality by using the integration operator , which is given by We will use the symbol instead of throughout the paper. Relevant information on the properties of can be found in [12–14] and [6, Ch. 3] and the references therein. Since for all the inclusion always holds, the integration map can be defined also as an operator ; we use the same symbol in this case for this operator. It must be said that the spaces represent in fact the class of all order continuous -convex Banach lattices with a weak unit (see [11, Prop. 2.4] or [6, Prop. 3.30]) that means that our results can be applied to a broad class of Banach spaces.

As we said in Introduction, duality and vector valued duality for the spaces are fundamental tools in this paper. Regarding duality, fix a Rybakov measure for . Due to the order continuity of , its dual space () allows an easy description; it coincides with its Köthe dual (or associate space) ; that is, , whereand the duality is given by Information about a precise description of can be found in [2, 7, 15–17]. It must be said here that and coincide only in very special situations, for instance, for being a scalar measure. We will write for the weak topology on .

Regarding vector valued duality relations between spaces, , the integration map defines the continuous bilinear map given by , , . Note that is both sides norming for and ; that is, for every , , and the same happens dually for the case of functions .

In this paper we will consider the topology of pointwise convergence of the integrals that is the locally convex topology defined by the seminorms , , . The topology of pointwise weak convergence of the integrals is defined by the seminorms , , , and . It is also a locally convex topology on . It is easy to see that the norm topology is finer than all the others, and and are finer than , although and are not comparable in general. An exhaustive analysis of the topology has been done recently and can be found in [18] (see also the references therein). The reader can find more information about it in [1, 6, 7, 11, 16, 19]. The following result establishes the basic relations between the quoted topologies.

Proposition 1 (See Proposition 1 in [18]). *Let . If is -compact then and coincide on bounded subsets of . Moreover, if and is -compact then the weak topology and coincide on bounded subsets of . Consequently, if , is -compact if and only if is reflexive and the weak topology and coincide on .*

In this paper we will make a local use of the duality defined by the integration bilinear map . For consider a subspace . We say that a subspace is an *-dual* for if is -norming for ; that is, the function gives an* equivalent* norm for . We write for such a space . In the same way, we say that a subspace of is -bidual of (with respect to -dual ) if and is -norming for . Notice that the inclusion is not necessary for to be -norming for . For instance, if is an order continuous Banach function space and is the vector measure given by , , then for the space generated by the function in is -norming for , and also the space generated by in is -norming for . However, is not included in . But note also that given , , and being norming, it can always be assumed that just by defining the new as the subspace of generated by We will use this example later.

We say that a triple of -dual spaces as above is an -dual system. We can define the topology over as the one induced by all the seminorms , , and the topology for given by the seminorms , . A quick look at the proof of Proposition in [18] shows that a local version of this result is also true, that is, a version of this result writing instead of and instead of , where is an -dual space.

Let us show some examples. A natural -dual space of is ; in this case, we write simply for the topology . However, an -dual space may be very small. For instance, if the integration map is isomorphism, then the subspace generated by is -dual for . Obviously, for every subspace , is -dual for .

Let us finish this section by defining a fundamental class of operators related to the summability of sequences with respect to the -topology. It generalizes the class considered in Lemma of [1] and in [7, Section ]. Theorem in [1] provides a Pietsch type domination/factorization theorem for this family of operators. The local version of this result becomes the main tool for the proof of our results.

*Definition 2. *Let , , be a subspace of and a Banach subspace of . Let be a Banach space. An operator is -summing if there is a constant such that, for any finite set of functions , Of course, the integration map is always -summing for all , . Indeed, if , then

#### 3. The Dvoretzky-Rogers Theorem for the -Summability

Throughout this section, , and are Banach spaces, is an -valued vector measure, is a subspace of , and is an -dual system. We will consider the following sequential properties associated with compactness with respect to the -topology.

*Definition 3. *An operator is -sequentially compact if every bounded sequence in has a subsequence such that is a Cauchy sequence for each .

*Definition 4. *An operator is -sequentially completely continuous if whenever is a bounded sequence such that for every .

If we assume that (we can always make big enough to have it), then is -sequentially continuous. In the classical summing operators theory it is well known that any summing operator is weakly compact. However, not every -summing operator is -sequentially compact. For instance, given a Banach function space , define , . Then is an isomorphism which is -summing but it is not -sequentially compact in general as in this case the norm topology and the topology coincide. Then is not -compact unless is finite dimensional. Let us see that, under some compactness assumptions, the -summing operators behave similarly as absolutely summing operators. We need first an easy lemma.

Lemma 5. *Let and let be a subspace of and an -norming subspace for . Consider a Banach space valued -summing operator . Then is -summing for each .*

* Proof. *Let be such that . Take a finite set of functions . Then where is the constant associated with the -summability of .

Theorem 6. *Let . Let be a -summing operator. The following statements hold. *(i)*If is -compact then is -sequentially completely continuous.*(ii)*If is -compact and is -compact then is completely continuous.*(iii)*Finally, if is -compact and is reflexive, then is also weakly compact.*

* Proof. *(i) We have that satisfies that, for every finite set ,Taking into account that is an -dual system, it can be shown as in the case of Pietsch’s Domination Theorem for -summing operators (see Lemma in [1] and make the obvious modifications) that there is measure on the compact space such thatThis easily gives that factorizes through the following scheme (see Theorem in [1]):