Locally Convex Spaces

The aim of this preparatory chapter is to collect, mainly for the purpose of later reference and in order to fix some notation, several basic facts on linear spaces of arbitrary dimension. From the beginning, we shall assume that the underlying scalar field is either the real or the complex number field, since this will be the only case needed in subsequent discussions.

In this chapter we start our investigations on general topological vector spaces by introducing the basic concepts and giving the standard descriptions of linear topologies by means of particular neighbourhood bases of the zero vector. This is followed by a brief discussion of boundedness and of continuity of linear forms in 2.3. In the sections 2.4–2.6 we consider projective topologies as a first general method to generate new linear topologies from given ones. The usual universal characterization of cartesian products is supplemented in 2.5 by a rather exceptional one which, however, will turn out to be a convenient and powerful tool in subsequent discussions.

Every topological vector space E carries a natural uniform structure (in the sense of N. Bourbaki [2], see also H.Schubert [1]) which is determined by all sets {(x,y) ϵ E × E|x — y ϵ U}, U running through any 0-basis in E. Consequently, topological vector spaces are open for application of the results of the general theory of uniform spaces.

The existence of inductive linear topologies and the explicit description of one of its 0-bases in terms of 0-bases of the given spaces form the main topics of the first section of this chapter. As particular cases, we consider quotient spaces, direct sums, and inductive limits of topological vector spaces. We conclude with an examination of strict inductive limits and derive their most important properties.

The main purpose of the present chapter is to provide an appropriate frame to prove the closed graph theorem and its relative, the open mapping theorem, in substantial generality. The classical version of the closed graph theorem is the result due to S. Banach which states that every closed linear map between complete metrizable tvs is automatically continuous.

To describe appropriately several phenomena which frequently appear when considering spaces derived from classical analysis, we are now going to introduce several subclasses of general tvs by taking into account some sort of convexity. More precisely, we will consider tvs admitting a 0-basis consisting of so-called r-convex sets, 0 < r ≤ 1. The most significant case occurs when r = 1, and it is in fact this case which supports a theory rich enough to handle properly the most important spaces originating from analysis. Actually, the remaining part of this book is essentially devoted to the study of such spaces which are known as locally convex spaces. Nevertheless, non-locally convex spaces having a 0-basis of r-convex sets for some 0 < r < 1 do occur; we shall in particular meet them when studying ideals of operators between Banach spaces in Chapter 19. For such reasons, and also since it does not require much extra work, we do not simply stick to the case r = 1.

The validity of the Hahn-Banach theorem to be proved in this chapter is what makes locally convex spaces superior to general topological vector spaces. It assures the existence of sufficiently many continuous linear forms to support a powerful duality theory which we shall discuss in subsequent chapters.

In a sense, the present chapter is the most fundamental one for the entire theory of locally convex spaces. Duality is what makes this theory powerful because it establishes a tool to translate a problem on the space (where it may appear to be difficult) into one concerning its linear forms (which may happen to be much easier to handle). Duality also admits the replacement of the original topology by simpler ones when dealing with problems involving boundedness, convexity, continuity, etc..

The topology of a Hausdorff lcs E is determined by knowing what the corresponding dual pairing 〈E,E’〉 and what the equicontinuous sets in E’ are. Closed equicontinuous sets are σ(E’, E)-compact, and many important results on is can be obtained by studying the collection of compact spaces obtained in this way. This will be the topic of the present chapter.

Local convergence of a sequence in an lcs E means that the sequence is contained and convergent in the normed space E _B associated with some disk B in E. In 10.1, we are going to study a corresponding concept based on an arbitrary homology. 10.2 is devoted to a corresponding notion of completeness, and in 10.3 we deal with some pecularities occurring when the homology under consideration is the equicontinuous compactology on the dual of an les. The corresponding notion of convergence for sequences, and filters, is what we understand by equicontinuous convergence.

Barrels have been introduced in 8.3 as polars of weakly bounded sets, and we know already that every closed, absolutely convex O-neighbourhood is a barrel. If the converse is also true, then the lcs under consideration is said to be barrelled. This is the case e.g. for Fréchet spaces and many other important lcs. Among the most important results on barrelled lcs to be proved in 11.1 we mention the Banach-Steinhaus theorem and Pták’s extension of the classical open mapping theorem. 11.2 is devoted to the larger class of quasi-barrelled lcs. In 11.3 we discuss the permanence properties of these spaces, with special emphasis on the problem of subspaces. Semi-reflexive and reflexive lcs are investigated in 11.4, and 11.5 is devoted to the study of semi-Montel and Montel spaces. 11.6 contains some of the most important facts on Fréchet-Montel spaces. In 11.7 we consider these concepts again for spaces of continuous functions and determine in particular the barrelledness character of C(X) for both, the compact-open and the pointwise topology, X being a completely regular space. Mainly for the purpose of appropriate calculation of the duals of spaces of integrable functions, we have included in 11.8 some elementary details on uniformly convex spaces. The chapter concludes with some fundamental facts on Hilbert-spaces.

In order to appropriately treat questions concerning metrizability and completeness of the strong dual of a Hausdorff lcs, we are going to discuss, in some detail, several weakenings of the notion of a (quasi-)barrelled lcs, by taking into account certain countability conditions.

The concept of a bornological lcs originates from the desire to have linear maps with values in another lcs continuous if they only carry bounded sets into bounded sets. Replacing herein the bounded sets of the domain space by Banach disks leads to the concept of an ultrabornological lcs. In a slightly more general setting, the study of such spaces is initiated in 13.1. In 13.2 we show that the bornological character of an lcs is determined on rather small bornologies, and we characterize bornological spaces by completeness properties of certain topologies on the dual. 13.3 is devoted to the study of associated bornological and ultrabornological spaces. If an lcs is (strictly) webbed, then so is the associated ultrabornological lcs, and this can be used to prove a convenient closed graph theorem for linear mappings from an ultrabornological lcs into a webbed lcs. In 13.4, we briefly consider associated bornological spaces of strong duals of metrizable lcs.

In this chapter we investigate bases and related objects in topological vector spaces and in particular in locally convex spaces. No attempt is made to be complete, we only treat some of the most important topics which will be needed in later discussions.

The present chapter is the first out of two which are devoted to the theory of topological tensor products of locally convex spaces as it was developed by A. Grothendieck [9] (see also [2] and [7]).

Whereas the projective tensor topology is, in a precise sense, the strongest “reasonable” topology on the tensor product E ⊗ F of two Hausdorff lcs E and F, the injective tensor topology, or ε-topology, which we are going to discuss now is the weakest “reasonable” topology on E ⊗ F. What is meant by “reasonable” will become clear in Chapter 19.

The present chapter is devoted to the study of some of the most important classes of continuous linear operators between locally convex spaces, and in particular between Banach spaces. It is important to note that all these classes do have the ideal property, so that they furnish basic examples for the general theory of ideals to be considered in greater detail in Chapter 19 below. We start our investigations in 17.1 by presenting some fundamental results on compact operators. In particular, the relationship to Schwartz topologies is emphasized. 17.2 is devoted to weakly compact operators; especially, the factorization theorem for weakly compact operators with values in a Fréchet space is proved. In 17.3, we introduce nuclear operators and prove their basic properties. By examining the dual of an £-tensor product, we are led in 17.4 to the so-called integral operators of Grothendieck. Spaces of nuclear and integral operators between Banach spaces can be made into Banach spaces in a canonical fashion. The norm for integral operators relates in a simple way to the trace for finite operators. This is the starting point for our investigations in 17.5, culminating in Dean’s proof of one of the most important results in recent functional analysis: the principle of local reflexivity for Banach spaces. Finally, in 17.6, we discuss some particular cases, such as Grothendieck’s results on the representability of operators defined on an L₁ (μ)-space and with values in a Banach space, and on the equivalence of nuclear and integral operators.

Roughly speaking, the approximation problem is the question if it is true, for a given lcs E, that every operator in L (E, E) can be approximated by finite rank operators, uniformly on compact sets. If E is a Banach space, then this is equivalent to asking whether every compact operator from any Banach space with values in E can be approximated by finite rank operators in the operator norm. The problem and most of the results in this area are due to A. Grothendieck [9]. But it was P. Enflo [1] who answered the problem in the negative through a fairly involved counter-example in 1973.

In a sense, the theory of ideals of operators in Banach spaces we are going to discuss now in some of its aspects can be looked at as an outgrow of part of A. Grothendieck’s work on topological tensor products. The setting is due to A. Pietsch (see[5]) and provides a powerful and convenient tool in both the local theory of Banach spaces and the study of particular classes of locally convex spaces. We shall deal with the latter topic in Chapter 21 below.

In this chapter we will present some additional results for components A(E, F) of an ideal A for particular Banach spaces E and F. Of major interest is the case where E and (or) F are Hilbert spaces, and we are going to discuss some aspects of such a situation.

Besides the class of Banach spaces, the nuclear locally convex spaces to be discussed in this chapter surely consitute one of the most important particular classes of locally convex spaces. In a specific sense, nuclear spaces are even closer to finite-dimensional spaces than Banach spaces are. In fact, they enjoy the Heine-Borel property (bounded sets are precompact), and, even more important, unconditional summability for sequences is equivalent to absolute summability.