This book has grown out of the Linear Analysis course given in Cambridge on numerous occasions for the third-year undergraduates reading mathematics. It is intended to be a fairly concise, yet readable and down-to-earth, introduction to functional analysis, with plenty of challenging exercises. In common with many authors, I have tried to write the kind of book that I would have liked to have learned from as an undergraduate. I am convinced that functional analysis is a particularly beautiful and elegant area of mathematics, and I have tried to convey my enthusiasm to the reader.

In most universities, the courses covering the contents of this book are given under the heading of Functional Analysis; the name Linear Analysis has been chosen to emphasize that most of the material in on linear functional analysis. Functional Analysis, in its wide sense, includes partial differential equations, stochastic theory and non-commutative harmonic analysis, but its core is the study of normed spaces, together with linear functionals and operators on them. That core is the principal topic of this volume.

Functional analysis was born around the turn of the century, and within a few years, after an amazing burst of development, it was a wellestablished major branch of mathematics. The early growth of functional analysis was based on 19th century Italian function theory, and was given a great impetus by the birth of Lebesgue's theory of integration.