This textbook provides a modern introduction to advanced concepts and methods of mathematical analysis.

The first three parts of the book cover functional analysis, harmonic analysis, and microlocal analysis. Each chapter is designed to provide readers with a solid understanding of fundamental concepts while guiding them through detailed proofs of significant theorems. These include the universal approximation property for artificial neural networks, Brouwer's domain invariance theorem, Nash's implicit function theorem, Calderón's reconstruction formula and wavelets, Wiener's Tauberian theorem, Hörmander's theorem of propagation of singularities, and proofs of many inequalities centered around the works of Hardy, Littlewood, and Sobolev. The final part of the book offers an overview of the analysis of partial differential equations. This vast subject is approached through a selection of major theorems such as the solution to Calderón's problem, De Giorgi's regularity theorem for elliptic equations, and the proof of a Strichartz-Bourgain estimate. Several renowned results are included in the numerous examples.

Based on courses given successively at the École Normale Supérieure in France (ENS Paris and ENS Paris-Saclay) and at Tsinghua University, the book is ideally suited for graduate courses in analysis and PDE. The prerequisites in topology and real analysis are conveniently recalled in the appendix.

Thomas Alazard is a senior researcher at CNRS and has taught at the École Normale Supérieure Paris, the École Normale Supérieure Paris-Saclay, and École Polytechnique. His research focuses on the analysis of partial differential equations.

Part I Functional Analysis.- 1 Topological Vector Spaces.- 2 Fixed Point Theorems.- 3 Hilbertian Analysis, Duality and Convexity.- Part II Harmonic Analysis.- 4 Fourier Series.- 5 Fourier Transform.- 6 Convolution.- 7 Sobolev Spaces.- 8 Harmonic Functions.- Part III Microlocal Analysis.- 9 Pseudo-Differential Operators.- 10 Symbolic Calculus.- 11 Hyperbolic Equations.- 12 Microlocal Singularities.- Part IV Analysis of Partial Differential Equations.- 13 The Calderón Problem.- 14 De Giorgi's Theorem.- 15 Schauder's Theorem.- 16 Dispersive Estimates.- Part V Recap and Solutions to the Exercises.- 17 Recap on General Topology.- 18 Inequalities in Lebesgue Spaces.- 19 Solutions.