Locally Convex Spaces: (269 Graduate Texts in Mathematics)

For most practicing analysts who use functional analysis, the restriction to Banach spaces seen in most real analysis graduate texts is not enough for their research. This graduate text, while focusing on locally convex topological vector spaces, is intended to cover most of the general theory needed for application to other areas of analysis. Normed vector spaces, Banach spaces, and Hilbert spaces are all examples of classes of locally convex spaces, which is why this is an important topic in functional analysis. While this graduate text focuses on what is needed for applications, it also shows the beauty of the subject and motivates the reader with exercises of varying difficulty. Key topics covered include point set topology, topological vector spaces, the Hahn–Banach theorem, seminorms and Fréchet spaces, uniform boundedness, and dual spaces. The prerequisite for this text is the Banach space theory typically taught in a beginning graduate real analysis course.

Table of Contents:
​​​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.- Bibliography.- Index.

About the Author :
M. Scott Osborne is currently Professor Emeritus of Mathematics at the University of Washington.

Review :
“I found much to enjoy and admire in this well-motivated, tightly organised introduction to the theory of locally convex spaces. It is a genuine graduate textbook, designed to be of maximum utility to those encountering this area of functional analysis for the first time.” (Nick Lord, The Mathematical Gazette, Vol. 99 (546), November, 2015) “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. … 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 additionaltext in a first course in functional analysis.” (Ittay Weiss, MAA Reviews, September, 2015) “The book presents an essential part of the general theory of locally convex spaces dealt with in functional analysis. … The book is well written, accessible for students and it contains a good selection of exercises.” (Enrique Jordá, Mathematical Reviews, August, 2014) “This is a great book about the set theory of real and complex numbers in addition to being a good reference on topological vector spaces. I recommend it to all logicians and philosophers of logic. It should appeal to abstract mathematicians, students at the undergraduate/ and graduate levels.” (Joseph J. Grenier, Amazon.com, August, 2014) “The book is well written, it is easy to read and should be useful for a one semester course. The proofs are clear and easy to follow and there are many exercises. The book presents in an accessible way the classical theory of locally convex spaces, and can be useful especiallyfor beginners interested in different areas of analysis … . a good addition to the literature on this topic.” (José Bonet, zbMATH, Vol. 1287, 2014)