**Ebook Info**

**Published:**2014**Number of pages:**227 pages**Format:**PDF**File Size:**9.55 MB**Authors:**Kenneth Hoffman

**Description**

A classic of pure mathematics, this advanced graduate-level text explores the intersection of functional analysis and analytic function theory. Close in spirit to abstract harmonic analysis, it is confined to Banach spaces of analytic functions in the unit disc.The author devotes the first four chapters to proofs of classical theorems on boundary values and boundary integral representations of analytic functions in the unit disc, including generalizations to Dirichlet algebras. The fifth chapter contains the factorization theory of Hp functions, a discussion of some partial extensions of the factorization, and a brief description of the classical approach to the theorems of the first five chapters. The remainder of the book addresses the structure of various Banach spaces and Banach algebras of analytic functions in the unit disc.Enhanced with 100 challenging exercises, a bibliography, and an index, this text belongs in the libraries of students, professional mathematicians, as well as anyone interested in a rigorous, high-level treatment of this topic.

**User’s Reviews**

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Kenneth Hoffman, a national treasure, just died of a heart attack at the age of 77. Banach Spaces of Analytic Functions is for math graduate students only. For people with only a college math major, I recommend Professor Hoffman’s wonderful book “Analysis in Euclidean Space”. His use there of the idea of a countable sub covering to show that the Heine Borel Theorem and the Bolzano Weierstrass Theorem are the same theorem is breathtaking. And the sophistication is matched by the sincerity. Somewhere he writes that if you are having trouble with a proof, the problem is usually that you have only a hazy understanding of the theorem. Go back and reread the theorem over and over and over again. Once you have freed yourself from all your inappropriate associations, the proof will be easy. Everything than Professor Hoffman wrote is pure gold. Banach Spaces of Analytic Functions should be on every mathematician’s shelf.

⭐Very well written book on Banach spaces and theirmany applications in mathematics. Highly recommendedfor graduate students working in the areas of puremathematics.

⭐A classic analysis book for you if you are interested in function spaces, analysis etc.

⭐good book

⭐This is a classical and nice book. Amazon service was pretty nice and shipped in time.

⭐The theory of Hardy spaces is vast, along with its applications. This book overviews what was known about them in the early 1960s. In spite of its age, it can still be read profitably by anyone interested in harmonic analysis and Hardy spaces. Chapter 1 gives a quick review of the mathematical background needed for reading the rest of the book, mostly dealing with measure theory, and Banach and Hilbert spaces. In chapter 2, the author gives a detailed treatment of Fourier series over the closed interval from -pi to pi. The chapter is designed to answer two questions, namely whether a function is determined by its Fourier series, and given a particular Fourier series, how one can recapture the function. These questions must be addressed in the appropriate norm on the Banach space of Lp spaces of Lebesgue integrable functions. There are many methods of recapturing the function, and the author discusses a few such methods, one being the Cesaro means. The authors proves that for a function in Lp, the Cesaro means of the Fourier series of the function converge to it in the Lp norm (when p is greater than or equal to 1 but less than infinity). When p is infinity, the author shows this is true in the weak-star topology. The author then shows how the Cesaro means can be used to characterize the different types of Fourier series. Analytic and harmonic functions in the unit disk are defined and studied in chapter 3. The first question the author addresses is to what extent these functions are determined by their boundary values. The author shows how to represent these functions on the closed unit disk using the Cauchy and Poisson integral formulas, thus answering this question. The second question he addresses is the behavior of these functions on the boundary, i.e. the Dirichlet problem. His methods for harmonic functions are analagous to those for Lp under the guise of Cesaro means, i.e. Cesaro summability becomes Abel summability. The author shows this connection more rigorously by proving Fatou’s theorem. Hp spaces are defined in this chapter, and the author illustrates one of the major differences between the harmonic and analytic functions. The author begins the study of H1 spaces in chapter 4, initially via the Helson-Lowdenslager approach. He first proves Fejer’s theorem for functions which are continuous on the closed unit disk and analytic at each interior point: the real parts of these functions are uniformly dense in the space of real-valued continuous functions on the unit circle. Szego’s theorem, which gives a measure of the “distance” from the constant function 1 to the subspace of these functions that vanish at the origin, is proved, as well as the Riesz theorem, which shows that analytic measures on the unit circle are absolutely continuous with respect to Lebesgue measure. He then applies these results to H1 functions, showing that such functions cannot vanish on a set of positive Lebesgue measure on the circle without being identically zero. The author then generalizes these results to Dirichlet algebras later in the chapter, showing to what extent the Riesz theorem carries over. The important factorization theorems for Hp functions are covered in chapter 5, wherein the famous Blaschke products come in. Their properties are discussed in detail, along with the ability to represent a non-zero bounded analytic function in the unit disk in terms of them. The author proves a theorem of Hardy and Littlewood on H1 functions of bounded variation and a theorem of Hardy on the growth of the Fourier coefficients of an H1 functions. The author studies the algebra A of continuous functions on the closed unit disk which are analytic on the open disk in chapter 6. The factorization results of chapter 5 are used along with the theory of commutative Banach algebras to characterize completely the closed ideals in A. Wermer’s maximality theorem, which states that A is a maximal closed subalgebra of the continuous complex-valued functions on the unit circle, is proven. The shift operator on the (Hilbert) space H2 is studied in chapter 7, the goal being to classify the invariant subspaces of this operator. The author uses a more classical approach due to Helson and Lowdenslager to do this. The shift operator on L2 (on the unit circle) is then considered, and its invariant subspaces described. The author finishes the chapter with a short discussion of the representations of H(infinity). After a study of Hp spaces on the half-plane in chapter 8, in chapters 9 and 10 the author predominantly looks at Hp and H(infinity) from a “soft” analysis point of view. He shows that the isometries of H1, induced by conformal mappings of the unit disk onto itself, can be studied by studying the isometries of H(infinity). The projections from Lp to Hp are discussed, the author providing readers the necessary background for a study of Toeplitz operators, if they so desire. The topology of the maximal ideal space of H(infinity) is considered, but at the time of publication it was not known whether or not the open unit disk is dense in this space. This is the famous corona theorem of Lennart Carleson, which he proved as this book went into publication.

⭐the book is a cornerstone of any serious inquiry in Hardy spaces and the invariant subspace problem; it is also hightly readable and well written. people interested in a second course on complex functions, harmonic analysis and functional analysis (banach and hilbert spaces) should have a look at it; it deserves it and the reader will be richly rewarded…

⭐”Polish” spaces are what follows Hilbert spaces, as night follows day. In a world where fractals and their functional analysis are everywhere, Banach spaces are necessary. Several years ago I reinvented Banach space in my ignorance while studying fractal theory. A book that is both cheap and gives a running shot at learning about this complex graduate level subject is important! I won’t say this is an easy book, but for the price it is well worth it as a doorway to a new world of analytical function theory.

⭐This is an astonishingly good book on analytic function theory in the disc. It gives a comprehensive overview of Hardy spaces, solves the invariant subspace problem for the unilateral shift, and gives the ideal theory of the disc algebra and H^infty. It would be remiss not to note, though, that many details in arguments are left to the reader and reading through the chapter alone constitutes a good set of exercises.

⭐ハーディ族について知りたいと思い購入した。色々な具体例が取り扱われていて勉強するのが楽しみです。

⭐Not found.

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