Real analysis (Q5890953)

This textbook covers measure and integration on the one hand and functional analysis and some Fourier analysis on the other. The text consists of 7~chapters the first of which (``Introduction and preliminaries'') presents metric spaces. In Chapter~2 (``A glimpse of measure and integration''), the author introduces the integral, starting from an a priori given measure space, and proves the usual convergence theorems.NEWLINENEWLINEChapter~3 (``Construction of measures'') explains how to actually obtain measure spaces from more primitive data; hence the Lebesgue measure is introduced. The reader should be cautioned that in the author's vernacular a measure on a set is what traditionally is called an outer measure, as opposed to a measure on a \(\sigma\)-algebra, which is a \(\sigma\)-additive set function. There is also an in-depth discussion of regularity for (outer) measures in this chapter.NEWLINENEWLINEChapter~4 (``Functions of real variables'') includes applications in the classical setting of functions on \(\mathbb{R}\) or \(\mathbb{R}^n\) and discusses for instance functions of bounded variation, absolutely continuous functions, a.e.\ differentiability of such functions (based on the idea of differentiation of measures), mollifiers, Fubini's theorem, coordinate transformations in multiple integrals (again based on the idea of differentiation of measures) and much more.NEWLINENEWLINEChapter~5 (``Basic principles of linear analysis'') contains the fundamental theorems of functional analysis: uniform boundedness, open mapping and closed graph theorems, Hahn-Banach extension and separation theorem, Hilbert space theory, Fourier series. In Chapter~6 (``\(L^p\)-spaces''), the author takes a closer look at Lebesgue spaces, proves the duality \((L^p)'=L^q\), and studies the Hardy-Littlewood maximal function. Further, the Sobolev spaces \(W^{k,p}\) (for integer~\(k\)) are introduced.NEWLINENEWLINEIn the final Chapter~7 (``Fourier integral and Sobolev space \(H^s\)''), the Fourier transform is defined on \(L^1\) and \(L^2\) and the Hilbert scale of Sobolev spaces \(H^s\), \(s\in \mathbb{R}\), is introduced. The last section presents applications of Fourier analysis to probability theory, notably the central limit theorem.NEWLINENEWLINEI found the writing of this book quite dense at times; many arguments are delegated to the exercises that are scattered throughout the text. Still, readers will appreciate the wealth of material covered in this nice monograph.