Functional Analysis, Harmonic Analysis, and Measure Theory

Below is a list of reviews of books on functional and harmonic analysis and measure theory

Functional Analysis, Sobolev Spaces and Partial Differential Equations
Brezis, H. (2011). ISBN 9780387709130.
Prerequisites: Normed vector and Complete metric spaces, Measure Theory, Multivariable Calculus.

Brezis’ book is a very nice book about Functional Analysis AND Partial Differential Equations from a Functional Analytic point of view. Requirements for it are quite light, but need to be VERY well mastered. This book contains mainly two distinct parts, the first one is about pure and applied Functional Analysis (Chapters 1 - 7). The second one (Chapters 8 - 10) deals with the construction of Sobolev spaces and few applications to linear Partial Differential Equations, with few examples. It contains A LOT of Exercises with their solutions. This book mainly deals with the linear theory of PDEs and question of regularity of solutions, and many tools introduced can be used in more general cases up to some nonlinear PDEs.

First chapters (1, 2 and 3) deal with topology in abstract normed vector spaces and are needed for the following chapters. Chapter 4 deals with functional analytic properties of Lp spaces. Chapter 5 & 6 are here to discuss specific properties of Hilbert spaces and some specific linear Operators acting on them, it includes Spectral Theorem for Operators with compact resolvent. Chapter 7 is an introduction to abstract vector valued ODEs, in the Hilbertian case, through semigroup theory, the idea here is to define exponential of some unbounded operators. Finally Chapter 8 and 9 are about construction of Lp-based Sobolev spaces and their properties, first in the one dimensional, then in the n-dimensional case. Their applications to solve linear (elliptic) PDEs in the L²-based case require tools introduced earlier in the first part (e.g. Chapter 5). In the same vibe, Chapter 10 is an application of Chapter 7 to few examples like the Heat and the Wave Equations with few comments.

Fourier Analysis: An Introduction (Princeton Lectures in Analysis)
Stein, E. M., Shakarchi, R. (2011). ISBN 9781400831234
Prerequisites: Multivariable calculus, basic real analysis/advanced calculus, linear algebra, and comfort with complex valued functions

A short introduction to Fourier analysis and its applications in various fields requiring very little prerequisites. Begins by motivating Fourier analysis as it might appear in solving physical problems like the wave equation or the heat equation, then covers Fourier series, Cesàro/Abel summability and Fejér’s theorem, convergence of Fourier series, and application of Fourier series in solving problems formulated in other areas of mathematics and also from analysis. Then the Fourier transform on the real numbers and its many consequences are covered, along with the Fourier transform in euclidean space as well as its application in physics and medicine. The last part of the book covers finite Fourier analysis which focuses on the development of the Fast Fourier transform as well as the theory of Fourier analysis on finite abelian groups which is used to prove Dirichlet’s theorem in number theory finally.

The text is very well written and contains diagrams which aid in understanding; furthermore, there is an appendix at the end for concepts in analysis and multivariate calculus. The examples for the application of Fourier analysis are chosen well and are varied. This text is probably not appropriate for anyone who has already learned measure theory or more advanced real analysis, as the text is designed to be a survey of Fourier analysis needing minimal requirements.