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2003 | Book

Topological vector spaces over a valued division ring Read chapter Read first chapter

Topological Vector Spaces

Chapters 1–5

Author: Nicolas Bourbaki

Publisher: Springer Berlin Heidelberg

Book Series : Elements of Mathematics

Included in: Professional Book Archive

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About this book

This is a softcover reprint of the English translation of 1987 of the second edition of Bourbaki's Espaces Vectoriels Topologiques (1981).
This [second edition] is a brand new book and completely supersedes the original version of nearly 30 years ago. But a lot of the material has been rearranged, rewritten, or replaced by a more up-to-date exposition, and a good deal of new material has been incorporated in this book, all reflecting the progress made in the field during the last three decades.
Table of Contents.
Chapter I: Topological vector spaces over a valued field.
Chapter II: Convex sets and locally convex spaces.
Chapter III: Spaces of continuous linear mappings.
Chapter IV: Duality in topological vector spaces.
Chapter V: Hilbert spaces (elementary theory).

Table of Contents

Frontmatter
Chapter I. Topological vector spaces over a valued division ring
Abstract
Definition 1. — Given a topological division ring K (GT, III, § 6.7) and a set E such that E has
1°
the structure of a left vector space on K;
 
2°
a topology compatible with the structure of the additive group of E (GT, III, §1.1) and satisfying in addition the following axiom:
 
(EVT) the mapping (λ, x) ↦ λx of K × E in E is continuous, then E is called a left topological vector space over (or on) K.
Nicolas Bourbaki
Chapter II. Convex sets and locally convex spaces
Abstract
In §§ 2 to 1 of this chapter, we shall be concerned only with vector spaces and affine spaces over the field of real numbersR, and when we speak of a vector space or an affine space without giving its division ring of scalars explicitly, then it is to be understood that this division ring is the fieldR. For vector spaces onC, see § 8.
Nicolas Bourbaki
Chapter III. Spaces of continuous linear mappings
Abstract
In this chapter, all the vector spaces under consideration are vector spaces over a fieldK, which may beRorC.
Nicolas Bourbaki
Chapter IV. Duality in topological vector spaces
Abstract
Throughout this chapter, all the vector spaces under consideration are vector spaces over afieldKwhich is eitherRorC.
Nicolas Bourbaki
Chapter V. Hilbertian spaces (elementary theory)
Abstract
Throughout this chapter, K denotes the fieldRor the fieldC.For every complex number ξ = α + iβ (α, β real), ξ denotes the conjugate α — iβ of ξ; in particular, we have ξ = ξif and only if ξ is real.
Nicolas Bourbaki
Backmatter
Metadata
Title
Topological Vector Spaces
Author
Nicolas Bourbaki
Copyright Year
2003
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-61715-7
Print ISBN
978-3-540-42338-6
DOI
https://doi.org/10.1007/978-3-642-61715-7