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## Über dieses Buch

This book provides an introduction to the theory of topological vector spaces, with a focus on locally convex spaces. It discusses topologies in dual pairs, culminating in the Mackey-Arens theorem, and also examines the properties of the weak topology on Banach spaces, for instance Banach’s theorem on weak*-closed subspaces on the dual of a Banach space (alias the Krein-Smulian theorem), the Eberlein-Smulian theorem, Krein’s theorem on the closed convex hull of weakly compact sets in a Banach space, and the Dunford-Pettis theorem characterising weak compactness in L1-spaces. Lastly, it addresses topics such as the locally convex final topology, with the application to test functions D(Ω) and the space of distributions, and the Krein-Milman theorem.

The book adopts an “economic” approach to interesting topics, and avoids exploring all the arising side topics. Written in a concise mathematical style, it is intended primarily for advanced graduate students with a background in elementary functional analysis, but is also useful as a reference text for established mathematicians.

## Inhaltsverzeichnis

### 1. Initial Topology, Topological Vector Spaces, Weak Topology

Abstract
The main objective of this chapter is to present the definition of topological vector spaces and to derive some fundamental properties. We will also introduce dual pairs of vector spaces and the weak topology. We start the chapter by briefly recalling concepts of topology and continuity, thereby also fixing notation.
Jürgen Voigt

### 2. Convexity, Separation Theorems, Locally Convex Spaces

Abstract
Locally convex spaces are introduced as topological vector spaces possessing a neighbourhood base of zero consisting of convex sets. It is shown that then the topology can also be defined by a set of semi-norms. In order to show this and other features, we first treat separation properties.
Jürgen Voigt

### 3. Polars, Bipolar Theorem, Polar Topologies

Abstract
In a dual pair 〈E, F〉 one wants to define topologies on E associated with collections of suitable subsets of F. (This generalises the definition of the norm topology on the dual E′ of a Banach space E, in this case for the dual pair 〈E′, E〉.) Such a collection $$\mathcal M$$ defines a ‘polar topology’ on E, where the corresponding neighbourhoods of zero in E are polars of the members of $$\mathcal M$$. Examples of such topologies are the weak topology and the strong topology. In the first part of the chapter we define polars and investigate some of their properties.
Jürgen Voigt

### 4. The Tikhonov and Alaoglu–Bourbaki Theorems

Abstract
The central result of this chapter is the Alaoglu–Bourbaki theorem: Polars of neighbourhoods of zero in a locally convex space E are σ(E′, E)-compact subsets of E′. As a consequence in a dual pair 〈E, F〉 one concludes that, for a locally convex topology τ on E with (E, τ) = F, one always has σ(E, F) ⊆ τ ⊆ μ(E, F), where μ(E, F) is the Mackey topology on E, corresponding to the collection of absolutely convex σ(F, E)-compact subsets of F. As a prerequisite we show Tikhonov’s theorem, and as a prerequisite to the proof of Tikhonov’s theorem we introduce filters describing convergence and continuity of mappings in topological spaces.
Jürgen Voigt

### 5. The Mackey–Arens Theorem

Abstract
The first objective is to complete the discussion concerning compatible topologies on E for a dual pair 〈E, F〉, by showing that (E, μ(E, F)) = F. In examples we discuss ‘compatibility’ for non-locally convex topologies.
Jürgen Voigt

### 6. Topologies on E″, Quasi-barrelled and Barrelled Spaces

Abstract
The topics of this chapter draw their motivation, with a locally convex space E, from two questions: find topologies on E″ such that the canonical mapping κ: E → E″ is continuous, and investigate properties of topologies on E ensuring that κ is continuous, if E″ is provided with the strong topology β(E″, E′). The first issue leads to the ‘natural topology’ on E″, the second leads to ‘quasi-barrelled’ spaces, and in particular, the answer to the second question motivates the investigation of further related properties of locally convex spaces.
Jürgen Voigt

### 7. Fréchet Spaces and DF-Spaces

Abstract
Besides Hilbert spaces and Banach spaces occurring as function spaces in analysis, an important role is also played by Fréchet spaces. It is for this reason that we include a chapter on some properties of metrisable locally convex spaces and Fréchet spaces. The first part of the chapter concerns the duality of Fréchet spaces: in short and simplified, duals of Fréchet spaces are DF-spaces, and duals of DF-spaces are Fréchet spaces. Looking at examples of duals of Fréchet spaces, one realises that quite often they can only be described as quotients, and this is the reason for inserting a short interlude on final topologies and topologies on quotient spaces. The third topic is a peculiarity of Fréchet spaces: They could also have been defined as ‘completely metrisable’ locally convex spaces.
Jürgen Voigt

### 8. Reflexivity

Abstract
We start by discussing semi-reflexivity and Montel spaces and present a number of examples of function spaces. At the end we present duality properties for reflexive spaces and Montel spaces.
Jürgen Voigt

### 9. Completeness

Abstract
Completeness is a property of a topological vector space as a ‘uniform space’. We do not explicitly use uniform spaces but mention that the linear structure allows to define neighbourhoods of ‘uniform size’ for all by taking the translates x + U for . This allows to introduce the notion of Cauchy filters, and completeness requires Cauchy filters to be convergent.
Jürgen Voigt

### 10. Locally Convex Final Topology, Topology of

Abstract
The topic of this chapter is of interest because of its applications to function spaces occurring in partial differential equations. In particular, we describe a neighbourhood base of zero for the space $$\mathcal D(\Omega )$$ of ‘test functions’.
Jürgen Voigt

### 11. Precompact – Compact – Complete

Abstract
This chapter is a short survey on the technical properties mentioned in the title, for subsets of topological vector spaces and locally convex spaces.
Jürgen Voigt

### 12. The Banach–Dieudonné and Krein–Šmulian Theorems

Abstract
In this and the following two chapters we discuss some surprising properties concerning the weak topology of Banach spaces. (However, the discussion will not be restricted to Banach spaces!)
Jürgen Voigt

### 13. The Eberlein–Šmulian and Eberlein–Grothendieck Theorems

Abstract
The well-known Eberlein–Šmulian theorem states the equivalence of several versions of weak compactness for subsets of a Banach space. The proof will be a consequence of properties of subsets H ⊆ C(X) with respect to the product topology on , where X is a suitable topological space. These considerations will also yield results for more general locally convex spaces.
Jürgen Voigt

### 14. Krein’s Theorem

Abstract
Another surprising result is Krein’s theorem, stating that the closed convex hull of a weakly compact set in a Banach space is again weakly compact. This will be shown in a much more general context. For the proof, the Pettis integral of vector-valued functions will be defined and applied.
Jürgen Voigt

### 15. Weakly Compact Sets in L 1(μ)

Abstract
In view of the discussion of properties of weakly compact sets in the last chapters, it seems appropriate to present examples of weakly compact sets in a non-reflexive space. Besides the characterisation of weak compactness of subsets of L 1(μ), we will also show that L 1(μ) is weakly sequentially complete.
Jürgen Voigt
Abstract
The issue of this chapter is to present an example where one can compute the bidual of a locally convex space without having an explicit description of the dual. This example could have been inserted much earlier, in fact after Chapter 8. We have preferred, however, to first pursue more theoretical developments.
Jürgen Voigt

### 17. The Krein–Milman Theorem

Abstract
The Krein–Milman theorem asserts that in a Hausdorff locally convex space all points of a compact convex set can be approximated by convex combinations of its ‘corners’. We show that this can be reinforced to the statement that all points of the set are barycentres of probability measures living on the closure of the extreme points of the set. An interesting application to completely monotone functions on [0, ) yields Bernstein’s theorem concerning Laplace transforms of finite Borel measures on [0, ).
Jürgen Voigt

### Backmatter

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