Teschl G. Topics in Real and Functional Analysis

Wien: Universitat Wien, 2018. — 570 p.This manuscript provides a brief introduction to Real and (linear and nonlinear) Functional Analysis. It covers basic Hilbert and Banach space theory as well as basic measure theory including Lebesgue spaces and the Fourier transform.Functional Analysis.
A first look at Banach and Hilbert spaces.
Hilbert spaces.
Compact operators.
The main theorems about Banach spaces.
Further topics on Banach spaces.
Bounded linear operators.
Operator semigroups.
Real Analysis.
Measures.
Integration.
The Lebesgue spaces Lp.
More measure theory.
The dual of Lp.
Sobolev spaces.
The Fourier transform.
Interpolation.
Nonlinear Functional Analysis
Analysis in Banach spaces.
The Brouwer mapping degree.
The Leray–Schauder mapping degree.
The stationary Navier–Stokes equation.
Monotone maps.
App. A. Some set theory.
App. B. Metric and topological spaces.Glossary of notation.