Analysis and Partial Differential Equations

<p>This textbook provides a modern introduction to advanced concepts and methods of mathematical analysis.</p><p>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 Calder��n's reconstruction formula and wavelets Wiener's Tauberian theorem H��rmander'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 Calder��n'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.</p><p>Based on courses given successively at the ��cole Normale Sup��rieure 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</p>