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A Primer on Hilbert Space Theory

Carlo Alabiso and Ittay Weiss
Publication Date: 
Number of Pages: 
UNITEXT for Physics
[Reviewed by
Mark Hunacek
, on

They say you can’t judge a book by its cover — apparently sometimes not by its title either: Hilbert spaces do not occupy the position of prominence in this book that its title would leave one to suspect they do, and in fact are not even formally defined until page 156, at which point there are only about sixty pages of text remaining in the book. Even in these sixty pages, a lot of attention is given to topics other than Hilbert spaces, and of the various topics that one does associate with Hilbert spaces per se, the only one that is discussed here is the Reisz representation theorem, which characterizes the dual space of a Hilbert space H. A better title for this book, therefore, would, I think, be something along the lines of “An Introduction to Functional Analysis”, because (with some exceptions) this text amounts to a gentle introduction to the basic theorems of that broader subject. It is, therefore, not so much a book on Hilbert spaces as it is a book designed to lay the groundwork for such a book. Readers who want a book more tightly focused on Hilbert spaces might wish to consult texts like Debnath’s Introduction to Hilbert Spaces with Applications and Young’s An Introduction to Hilbert Space.

The late introduction of Hilbert spaces is due to the authors’ attempt to make this book as broadly accessible as possible by including a lot of prefatory material. As a result, prerequisites are kept to a minimum; good courses in linear algebra and real analysis should suffice as background. Measure theory and the Lebesgue integral are not used. The spaces \(L_p[a,b]\) are defined as the completion of the space \(C[a,b]\) with a \(p\)-power integral norm defined by the Riemann integral.

A necessary consequence of the fact that measure theory is not used is that some results about \(L_p\) spaces (e.g., the characterization of their dual spaces) are not proved here. For these reasons, although the text is advertised as being suitable for advanced undergraduates or early graduate students, many graduate mathematics instructors may conclude this book is too elementary for their purposes, although instructors of graduate courses in other areas, like physics, might find this book suitable.

The text begins pretty much at the beginning, with a discussion of set theory, including cardinal number arithmetic and Zorn’s lemma. The next chapter is on linear algebra (linear spaces, operators, inner product spaces) with, of course, no assumption of finite-dimensionality made; proofs of standard linear algebra results (e.g., the existence of a basis) are given for arbitrary vector spaces.

This is followed by two chapters on topological and metric spaces, respectively. These chapters comprise about 75 pages and summarize many of the basic results that one would expect to find in a beginning course on topology, although some results, such as Tychonoff’s theorem for arbitrary products, are stated without proof (the proof for finite products is an exercise), and other topics are treated in a rather succinct way; for example, quotient spaces are defined, and a simple example (\(S^1\)as a quotient space of \([0, 1]\)) given, in two paragraphs. Likewise, while it is proved that any path-connected space is connected, the statement of that result is not followed, as one might expect it to be, by a discussion of the converse.

Continuing with the trend of discussing increasingly more specific objects, the next chapter mostly discusses normed spaces, although semi-normed spaces are also mentioned. It is in this chapter that Banach and Hilbert spaces are defined and their basic properties explored, including proofs of the main theorems of functional analysis (Hahn-Banach, Open Mapping, and Closed Graph) and the determination of the dual spaces of the sequence spaces \(\ell_p\). In a nice touch, there is also a good discussion of functional-analytic techniques in the study of integral equations. The chapter ends with a quick look at unbounded operators.

The final chapter of text is on topological groups, and since it assumes no prior knowledge of group theory, it starts with the definition of a group. The relationship between topological groups and the previously discussed material appears in this text largely from a consideration of the few examples given; for example, the group of invertible bounded linear operators of a Banach space is shown to be a topological group. There is also a section in this chapter on uniform spaces, a topic not generally discussed at this elementary level.

Each of these six chapters ends with exercises, most of which struck me as pretty standard and not terribly difficult. No solutions are provided, but there is also a chapter 7, which presents an assortment of solved problems in all the areas of the six preceding chapters. Sometimes these problems are used as a way of establishing some well-known fact, such as the path-connectedness of the special orthogonal group \(\mathrm{SO}(n)\).

One of the two coauthors of this book (Alabiso) is a physicist and the other (Weiss) is a mathematician. In view of this interesting collaboration, I was a bit disappointed to see very little discussion of the connection between Hilbert spaces and physics — such as, say, the use of Hilbert space in quantum mechanics. (Packel’s Functional Analysis and Saxe’s Beginning Functional Analysis, for example, show that it is possible to give a brief but satisfying look at this topic.) Likewise, the lack of any discussion of Fourier series also seems like a missed opportunity.

Although the choice of topics struck me as somewhat idiosyncratic, I did feel that what the book does, it does quite nicely. The authors’ writing style is clear, and, as previously noted, they take pains to make this book broadly accessible. Each chapter begins, for example, with an abstract setting out what is to be done in the chapter, providing a list of “key words”, and giving some sense of perspective. So this book, or at least large portions of it, could likely be read by advanced undergraduates — although whether there is much demand for this material at the undergraduate level is certainly a question. Anybody who has worked their way through this book should be in a good position to tackle more demanding texts, either on Hilbert spaces or other topics in functional analysis.

Mark Hunacek ( teaches mathematics at Iowa State University.