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Locally Convex Spaces

M. Scott Osborne
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 269
[Reviewed by
Ittay Weiss
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Locally convex spaces are the unavoidable creatures one must learn to handle when the theory of Banach spaces gets too rigid for applications. As the author of this very friendly book explains, most texts on functional analysis either restrict attention to Banach spaces or cover the general theory in great detail. The aim of the book is to explore the theory of locally convex spaces relying only on a modest familiarity with Banach spaces, and taking an applications oriented approach. This allows the author to touch upon subjects and results that are not typically found in other texts.

The book is very well suited to graduate students. It starts off rather leisurely with a chapter on topological groups and a chapter on topological vector spaces, setting the stage for locally convex spaces, which appear in the following chapter. The treatment is precise, drawing on the assumed familiarity with Banach spaces, and, very importantly, allowing the inherent beauty of the subject matter to flow easily from the text. The classical results, namely the open mapping theorem and the closed graph theorem, form the topic of the next chapter, with the remaining two chapters devoted to dual spaces. These latter chapters set the book apart from other texts by exploring important results that are not in the typical range of topics in beginner functional analysis texts.

With plenty of exercises accompanying the text, an appendix with hints for selected exercises, and three more appendices touching upon theory slightly outside of the immediate scope of the book, the author’s very focused aim and clear exposition makes the book an excellent addition to the literature. The book is suitable for self-study as well as a textbook for a graduate course. The book can also be prescribed as additional text in a first course in functional analysis.

Ittay Weiss is Lecturer of Mathematics at the School of Computing, Information and Mathematical Sciences of the University of the South Pacific in Suva, Fiji.

​​​1 Topological Groups

2 Topological Vector Spaces

3 Locally Convex Spaces

4 The Classics

5 Dual Spaces

6 Duals of Fré​chet Spaces

A Topological Oddities

B Closed Graphs in Topological Groups

C The Other Krein–Smulian Theorem

D Further Hints for Selected Exercises