Skip to main content

2017 | Buch

Topological Vector Spaces and Their Applications

insite
SUCHEN

Über dieses Buch

This book gives a compact exposition of the fundamentals of the theory of locally convex topological vector spaces. Furthermore it contains a survey of the most important results of a more subtle nature, which cannot be regarded as basic, but knowledge which is useful for understanding applications. Finally, the book explores some of such applications connected with differential calculus and measure theory in infinite-dimensional spaces. These applications are a central aspect of the book, which is why it is different from the wide range of existing texts on topological vector spaces. Overall, this book develops differential and integral calculus on infinite-dimensional locally convex spaces by using methods and techniques of the theory of locally convex spaces.

The target readership includes mathematicians and physicists whose research is related to infinite-dimensional analysis.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction to the theory of topological vector spaces
Abstract
In this chapter we present basic concepts and examples connected with topological vector spaces.
V. I. Bogachev, O. G. Smolyanov
Chapter 2. Methods of constructing topological vector spaces
Abstract
In this chapter we consider projective limits (in particular, products) of families of topological vector spaces, inductive limits (in particular, topological direct sums) of families of locally convex spaces, including strict inductive limits and inductive limits with compact embeddings, tensor products of locally convex spaces, and nuclear spaces.
V. I. Bogachev, O. G. Smolyanov
Chapter 3. Duality
Abstract
A powerful method of proving a great number of results in the theory of locally convex spaces employs passage from some subsets in such spaces to their polars, which are subsets of the dual spaces. Moreover, in place of properties of the original sets certain properties of their polars are studied and then one returns back, more precisely, to the polars of polars (the so-called bipolars), which are absolutely convex closed hulls of the original sets.
V. I. Bogachev, O. G. Smolyanov
Chapter 4. Differential calculus
Abstract
The concept of differentiable mapping from a topological vector space to a topological vector space was worked out relatively recently. In the mid 60s of the XX century the number of existing definitions of differentiability of mappings of topological vector spaces was very large and was comparable (if not greater) with the number of papers devoted to the study of such mappings.
V. I. Bogachev, O. G. Smolyanov
Chapter 5. Measures on linear spaces
Abstract
In this chapter we give a brief account of measure theory on linear spaces. We assume some acquaintance with basics of the Lebesgue theory of measure and integral (see, for example, Chapters 2 and 3 in [72]). We present the fundamental facts of the theory of Gaussian measures, discuss weak convergence of measures and the Fourier transform of measures.
V. I. Bogachev, O. G. Smolyanov
Backmatter
Metadaten
Titel
Topological Vector Spaces and Their Applications
verfasst von
V.I. Bogachev
O.G. Smolyanov
Copyright-Jahr
2017
Electronic ISBN
978-3-319-57117-1
Print ISBN
978-3-319-57116-4
DOI
https://doi.org/10.1007/978-3-319-57117-1