Philippe G. Ciarlet
SIAM 2013, 832 PAGES
PRICE (HARDBACK) £65.00 ISBN 978-1-611-97258-0
This book is a self-contained volume containing the fundamentals of linear and nonlinear functional analysis. It consists of nine chapters. The first is a very brief description of real analysis and functions of several real variables, it consists of theorems, no proofs are given. Chapters 2–5 cover all of linear functional analysis, Chapter 6 applies this knowledge to linear PDE, and Chapters 7–9 cover nonlinear functional analysis.
A description of the chapters follows.
Chapter 1 – Real Analysis and Theory of Functions: A Quick Review – This chapter is a very brief description of those topics that are essentially pre-requisites for reading this book. It covers basic real analysis, Lebesgue measure and integral in N-dimensional real spaces. It also covers basic m-differentiable function spaces C. Assumed knowledge includes first year undergraduate algebra, partial differentiation of real functions and Taylor’s formula. There are no proofs or exercises in this chapter.
Chapter 2 – Normed Vector Spaces – the start of linear functional analysis. This chapter covers continuous linear or multilinear operators, compact linear operators, compactness, approximation of functions in Lp by smooth functions. Using mollifiers is treated in a highly detailed fashion.
Chapter 3 – Banach Spaces – complete normed vector spaces. This chapter introduces various examples, the lp spaces and the Lp spaces of functions. Convergence is discussed. The Banach fixed point theorem is covered here, along with the Ascoli-Arzelà theorem.
Chapter 4 – Inner-Product Spaces and Hilbert Spaces – these spaces are very attractive to study since many of the properties of the Euclidean norm carry over to their norm. Amongst the material covered in this chapter are the Riesz representation theory and the spectral theory for compact self-adjoint operators.
Chapter 5 – The ‘Great Theorems’ of Linear Functional Analysis – these include Baire’s theorem, the Hahn-Banach theorem, the uniform boundedness principle, the Banach open mapping theorem and Banach’s closed graph theorem. Many consequences are discussed.
Chapter 6 – Linear Partial Differential Equations – consideration is limited to time independent problems. Applications include optimisation theory, linearised elasticity and linearised fluid mechanics. These are minimisation problems.
Chapter 7 – Differential Calculus in Normed Vector Spaces – this is the start of nonlinear functional analysis, which focuses on the idea of derivability of mappings between arbitrary normed vector spaces. This chapter covers the following: the Fréchet derivative, the chain rule, the mean value theorem in various forms, Sard’s lemma and the Schwarz lemma. The Piola identity is proved. Applications covered include analysis of necessary and sufficient conditions for extrema of real-valued functions.
Chapter 8 – Differential Geometry in Rn – this covers a review of basic differential geometry, e.g. the metric tensor and some tensor analysis. The fundamental theorem of Riemannian geometry is proved in detail. The two fundamental forms are reviewed – Gaussian curvature and covariant derivatives. The chapter closes with a detailed proof of the fundamental theorem of surface theory.
Chapter 9 – The ‘Great Theorems’ of Nonlinear Functional Analysis – this chapter aims to cover only the most basic of notions of nonlinear functional analysis as the subject is described as vast. First is an introduction to the calculus of variations, covering minimisation problems for nonquadratic functionals, usually defined in a Sobolev space. The solutions of these minimisation problems are the solutions of nonlinear partial differential equations. Existence theorems are then covered for sequentially weakly semicontinuous functionals. The second part of the chapter covers Brouwer’s fixed point theorem. A number of consequences of the fixed point theorem follow. Finally, Brouwer’s topological degree construction is seen in detail and used to prove some spectacular results.
This is a very large (800 page), intense book. The simpler parts of the book (linear functional analysis) are intended for at least final year undergraduate. Most of the contents appear to be aimed at PhD level.
The book is well laid out and many topics are covered. I would recommend the book for the more mature student of analysis.
John Bartlett CMath MIMA
Book review published directly onto IMA website (August 2015)
