Measure theory and functional analysis (Q2847908)

This book provides an introduction to measure theory and functional analysis suitable for a beginning graduate course, and is based on notes the author has developed over several years of teaching such a course. It is unique in placing special emphasis on the separable setting, which allows for a simultaneously more detailed and more elementary exposition, and for its rapid progression into advanced topics in the spectral theory of families of self-adjoint operators. The author's notion of measurable Hilbert bundles is used to give the spectral theorem a particularly elegant formulation not to be found in other textbooks on the subject.NEWLINENEWLINENEWLINEThe book consists of five chapters, each one containing a list of exercises. Chapter~ 1 deals with the main topological results needed for later purposes, especially in the context of separable metrizable spaces. In Chapter~ 2 Lebesgue integration is considered, and the most relevant theorems are proved in detail (e.g. monotone and dominated convergence theorems, or the Radon-Nikodym theorem). Banach spaces theory is the subject of Chapter~ 3, and special emphasis is made for the discrete sequence spaces \(\ell^1\), \(\ell^2\), \(\ell^\infty\) and \(c_0\). The Hahn-Banach theorem is proved for separable spaces (avoiding thus the use of the axiom of choice). The space \(C(X)\) (for a compact Hausdorff space \(X\)) is thoroughly studied in the final sections: Stone-Weierstrass theorem, Tietze extension theorem, etc. Duality theory and weak* topologies are the main topics in Chapter~ 4, starting with the Banach-Alaoglu theorem, the Krein-Smulian theorem, the Krein-Milman theorem and the Riesz-Makarov theorem characterizing the dual of \(C_0(X)\) (for a second countable locally compact Hausdorff space \(X\)). Duality of the \(L^p\) spaces, \(1\leq p<\infty\), is described in the last sections. Finally, Chapter~ 5 presents the main results on (separable) Hilbert spaces \(H\), like the Riesz representation theorem, the continuous functional calculus or the spectral theorems for bounded self-adjoint operators on \(H\).