Functional analysis. An introduction to Banach space theory (Q2784318)

The book under review is an introduction to the theory of Banach spaces. Still, many classical topics in functional analysis, like distributions, are not even mentioned. Locally convex spaces are introduced only to present the weak topologies on Banach spaces and, in general, there are no statements about these spaces. Some remarks in the introduction like ``the material in the book plays an integral role that every young analyst should know and master'' are a little exaggerated, since there are many (even old) analysts who do not know very much of certain specialized Banach theory theorems treated in the book under review.NEWLINENEWLINENEWLINEThe author should have pointed out some book (or books) as the background that someone who tries to study his book is supposed to have. Some theorem like the Baire category theorem, the Hausdorff maximality principle, Zorn's Lemma, etc. are used without any comment. Therefore, they should be known by the reader. The book also lacks a guideline on the interdependence of its chapters. This would have been helpful, since in some places the material treated in the book is rather specialized.NEWLINENEWLINENEWLINEAlthough, as the author states, the book is designed for a first two-semester course, it lacks exercises. It does not seem enough to repeat sentences like ``\dots as the student should verify\dots''. In many instances the book also lacks concrete examples.NEWLINENEWLINENEWLINEIf the book were entitled just ``An introduction to Banach space theory'', the selection of topics treated in the book under review would be quite good. In addition, besides some historical notes the author has done some interesting effort to motivate the student. A good feature is that some topics treated in the book under review, like strictly singular operators, are not easily found in the literature.NEWLINENEWLINENEWLINEThere are some errors, for instance, on page 4 the intersection and the union of sets are defined in the same way. In the introduction the author claims that \(\theta\) will denote the zero vector and that \(0\) will stand for the zero scalar. But then on page 61, when treating the inner product on Hilbert space, it can be seen \(\langle\theta, z\rangle= \theta\). In other places of the book the zero vector \(\theta\) is also denoted by \(0\). There are other problems with the notation. For instance, to say that a reflexive normed linear space is always a Banach space, because it is isometrically isomorphic to the dual space which is, in turn, a Banach space, the author writes a sentence as the following one (page 75): ``\dots you should notice that it follows readily from the isometric nature of the natural injection and the Banach space nature of dual spaces of normed linear spaces (that is, every dual space of a normed linear space is itself a Banach space being exactly \({\mathcal L}(X,\mathbb{R})\)), that reflexive normed linear spaces are always Banach spaces''. This kind of thing happens in many places and sometimes one can get confused. Indeed, the explanation of easy facts becomes a little cumbersome in many places.NEWLINENEWLINENEWLINEBut the biggest problem with the book under review is that sometimes one is not quite sure whether the author is writing about real Banach spaces or real and complex Banach spaces. It seems that only when in the statement of the theorems appear ``real Banach spaces'' it is referring to real Banach spaces. But in some places, for instance, on page 158 the author writes \(\{x^*(x): x\in B\}\) is bounded in \(\mathbb{R}\), yet does not seem to be talking about the real case. Another instance of this fact is that the author only states and proves the \(\ell^1\)-Rosenthal's Theorem (the real case) but not the complex version due to Dor. However, in Chapter 6 he uses the theorem for (apparently) any real or complex Banach space without any comment.NEWLINENEWLINENEWLINEThe book consists of an introduction plus six chapters.NEWLINENEWLINENEWLINEIn the introduction, some notation and conventions (some of them not very standard) are introduced. The product topology is also reviewed. Then finite-dimensional spaces and Riesz's lemma are studied. The author gives a review of the Daniell integral (this should have been better placed as an appendix).NEWLINENEWLINENEWLINEChapter 1 is devoted to presenting many classical examples of Banach spaces (even James's space is introduced). The dual spaces of many Banach spaces is also computed, including dual spaces of the sequence space \(\ell^1\) or the measure space \(L^1(\mu)\).NEWLINENEWLINENEWLINEIn Chapter 2, the author examines the basic theorems in functional analysis (in the Banach space setting) like Hahn-Banach, the Banach-Steinhaus theorem, the open mapping theorem and the closed-graphed theorem. The chapter concludes with some applications, Banach limits are introduced. Also some theory of infinite matrices is presented, like for instance, the Silverman-Toeplitz summability theorem and the Knopp-Lorentz summability theorem. The chapter ends with Helly's theorem about solutions of equations.NEWLINENEWLINENEWLINEChapter 3 begins with the study of the Minkowski functionals. Then the author presents the dual systems and weak topologies. The Banach-Alaoglu theorem, Goldstine's theorem, the Eberlein-Smulian theorem. The Krein-Milman theorem (for convex spaces) and the Stone-Weierstrass theorem are presented. No examples (other than those coming from the weak topologies) of locally convex spaces are given in this chapter. Few examples (if any) of Banach algebras are given.NEWLINENEWLINENEWLINEChapter 4 begins with the study of linear projections and adjoint operators. Then the author turns his attention to several classes of operators in Banach spaces: weakly compact operators; compact operators (including the Riesz-Schauder theory); strictly singular and cosingular operators. It is also proved that weakly compact operators, compact operators and strictly singular operators are Pietsch's ideals. The chapter ends studying the relation between reflexivity and factorization of weakly compact operators.NEWLINENEWLINENEWLINEChapter 5 is dedicated to the studying of bases in Banach spaces. The author presents the Schauder system for \({\mathcal C}([0,1])\) and the Haar system for the \(L^p\) spaces. Then it is also discussed the relation between equivalent basis and complemented subspaces. The selection principles of Bessaga and Pełcyński and the \(\ell^1\)-Rosenthal theorem are proved.NEWLINENEWLINENEWLINEFinally, Chapter 6 explores different subjects. It begins with Philip's lemma and some of its consequences, like Philip's theorem that asserts that \(c_0\) is not complemented in \(\ell^\infty\) and Schur's theorem that asserts that norm convergence and weak convergence are equivalent in \(\ell^1\). Then Dunford-Pettis operators are studied. Convex sets in Banach spaces and Bishop-Phelps theorem about subreflexibility of every Banach space are presented. The chapter finishes studying the convexity in Banach spaces and its relationship with the differentiation of the norm in Banach spaces.NEWLINENEWLINENEWLINEThe book could be recommended for those who have already a good knowledge in Banach space theory and are interested in some of the more special topics treated in the book.