From Vector Spaces to Function Spaces: Introduction to Functional Analysis with Applications


Yutaka Yamamoto
SIAM 2012, 260 PAGES
PRICE (HARDBACK) £63.00 ISBN 978-1-611-97230-6

From Vector Spaces to Function Spaces Introduction to Functional Analysis with ApplicationsThis book is aimed at almost a beginning undergraduate in Mathematics as it starts with the simplest view of a vector and proceeds to introduce the axioms of a vector space and all related concepts, as usually seen in a first year Algebra course. The book is probably easier for a second year to understand, since these ideas will then be well assimilated, however a well-motivated first year could find the book very interesting.

Subsequent chapters introduce Banach spaces, dual spaces that are related to continuous functionals, various aspects of linear mappings and their related operators. Chapter 6 provides a very good introduction to distribution space, Chapters 7 and 8 continue by applying distribution spaces to Fourier and Laplace transforms respectively. Chapter 9 provides some details about Hardy Spaces and from this point on the academic maturity of the book rises to some degree. Chapter 10 introduces Control and System Theory, which has been the reason for covering the various topics from earlier in the book.

As stated previously, the level of the earlier parts of the book lend it to use, certainly, by second year undergraduates of pure mathematics. The presence of many exercises in each chapter is another strong positive for the book. Chapters 9 and 10 are somewhat more challenging as the quantity of material covered would otherwise make the book too large.

John Bartlett CMath MIMA

Book review published directly onto IMA website (August 2014)

Published