## Hans Hamburger and Margaret Grimshaw: *Hilbert Spaces*

In 1951 Hans Ludwig Hamburger and Margaret Eleanor Grimshaw's book

*Linear Transformations in n-Dimensional Vector Space. An Introduction to the Theory of Hilbert Space*was published by Cambridge University Press. They had done most of the work for the book while colleagues at Southampton during the 1940s. We present below some extracts from reviews of this book.**1. Review by: Paul Richard Halmos.**

*Mathematical Reviews*, MR0041355

**(12,836b)**.

As the title and subtitle of this tract indicate, the material it covers is well known and can be found in many books. The authors manage, nevertheless, to present several facts, developments, and points of view that it would be difficult for a student to find in, or deduce from, the literature easily available to him. The novelty is achieved by a deliberate change of emphasis, opposed to some of the modern tendencies. As is proper in a treatment whose motivation is the infinite-dimensional case, none of the proofs relies on the theory of determinants. ... The authors have chosen to formulate the basic definitions in terms of coordinates. ... The effects of this decision are visible throughout and are seen most clearly in the fifth and last chapter (Vector spaces with positive Hermitian metric forms). Since the concepts of vector space and unitary space have not been defined before, it becomes necessary at this point to start all over again and to re-define and re-examine such concepts as linear manifold and Hermitian transformation. The quality of the book is such that every student of the subject, whether he agrees or disagrees with the point of view of the authors, will find it a valuable addition to his library.

**2. Review by: Mahlon Marsh Day.**

*Bull. Amer. Math. Soc.*

**59**(1) (1953), 98-99.

Since the purpose of this book is clearly indicated by its title and subtitle, this review can consider first the order (quite reasonable) and clarity (good) with which the authors present their material. ... Interesting historical notes, some referring to work of a century ago, show the authors' knowledge of the deep roots of their subject in the structure of classical mathematics. That their presentation is faithful to that same classical tradition may make the book easier for a student to begin, but it seems to this reviewer to make the secondary goal, the introduction of the reader to Hilbert space, so much the more difficult to reach in a small book. This reviewer can (as the pre-publication reviewer for Mathematical Reviews could not), and therefore should, attempt to compare this book with Finite dimensional vector spaces by P R Halmos. The books overlap much more in subject matter than in attitude; Halmos acknowledges great indebtedness to von Neumann, whose name does not appear in bibliography or index of the book under review. A student unaccustomed, as so many of our undergraduates are, to axiomatic methods might profit more from this concrete and detailed study than from a surfeit of abstractions. On the other hand, Halmos's book, with its racy style and its steady slant toward Hilbert space, when contrasted with this formal, workmanlike, and detailed discussion, seems to offer one of the few examples of a paperbound book suited better than its slick-paper competitor to the education of any student who has been bent to the appropriate axiomatic attitude.

**3. Review by: Jacob Lionel Bakst Cooper.**

*The Mathematical Gazette*

**37**(320) (1953), 156-157.

This book gives a very readable account of the theory, particularly of the reduction to canonical forms, of linear operators in

*n*-dimensional linear spaces over the complex field. As the title indicates, it is intended to act as an introduction to the theory of operators in Hilbert Space, and hence the authors have, on the whole, confined themselves to the use of methods which can be carried over to Hilbert Space. The essence of these methods is that they are as far as possible geometrical, operating with subspaces and linear operators rather than with determinants and with the explicit matrix expression of the linear operators. In this they resemble the methods which are finding favour with modern algebraists: and they could be carried over with little change to discussion of linear spaces over an arbitrary algebraically complete field, though they would not be suitable for a discussion of, say, the reduction of real matrices to canonical forms by real transformations. ... The authors have not allowed themselves to be bound too firmly by the intention expressed by their subtitle: they have at times made certain compromises with the evident aim of producing a richer and more readable account of*n*-dimensional space. This is seen most clearly in the third chapter: the variational methods there used cannot be applied to all Hilbert Space operators; but their use has the advantage of introducing the reader to ideas which are of great practical importance wherever they can be applied. More debatable points are the use made of the scalar product, and the definition of the space as a concrete space of sets of*n*numbers. The use of the scalar product has great technical advantages in calculating with subspaces, and is valuable because of its importance in Hilbert space; but its use here may disguise the fact that the theory expounded is true in affine, not merely in metric spaces. This point could easily be cleared up by explaining that an*n*-dimensional affine space is isomorphic with its dual. The definition of the vector space, and of the scalar product, by abstract arguments would have been a good introduction to the abstract definition of Hilbert Space, and would have saved a certain amount of repetition in the last chapter; however, the line the authors have chosen has advantages for less mature readers, and the treatment does not make much use of the coordinates. It can be said without hesitation that this is an excellently written account of its subject, and one that can confidently be recommended. We look forward to the authors' work on Hilbert Space.**4. Review by: Michael Golomb.**

*Science, New Series*

**115**(2990) (1952), 425.

The book by Hamburger and Grimshaw is a valuable addition to this steadily growing branch of mathematical literature. It introduces the ideas and methods of the theory of linear transformations in Hilbert space (the "operators" of quantum mechanics) by using them to present the more elementary theory in a finite dimensional space. Its principal results - the spectral representations of Hermitian transformations and the canonical representations and commutativity properties of general linear transformations - are developed in great detail. No use is made of determinants since there would be no analogous treatment in Hilbert space. Neither the material nor its treatment is new. Nevertheless, even the specialist in this field will find novelty in some results and proofs, and in the organization of the material. Of special interest are the instructive notes, mostly historical and bibliographical, and the copious references. To the non-specialist the book will not be easy reading. Although it is intended as an introduction to a more advanced theory, the authors pay little attention to didactical demands. Motivation of developments is brief or nonexistent, there is practically no illustrative material, no excursions to other fields, no signposts to guide us through the more complicated proofs. The authors state in the preface that they chose, for the greater part of the book, a concrete vector space in preference to an abstract space, so that the ideas may be more readily grasped. But not only does this choice necessitate duplication and circumlocution, but, by stressing the arithmetical against the geometrical aspect, it may actually produce the opposite effect. Whatever objections one may have to these features of the book, its positive qualities are such that it deserves the widest use.

**5. Review by: Richard Vincent Kadison.**

*Amer. Math. Monthly*

**59**(9) (1952), 650-652.

This book has a certain amount to recommend it. There is a wealth of material to be found in it which is usually not contained in an elementary text. The eigenvalue inequalities, the physical applications, and the analytical techniques are but a few examples of this. Some of the more powerful methods of modern functional analysis are illustrated in the simplified finite dimensional situation in such a manner as to make them readily accessible to the beginning student. On the other hand, the reviewer found many objectionable features to this book. The notation is made clumsy by using superscripts on letters to denote vectors in a set of vectors and subscripts on letters for complex numbers in a set. The non-invariant concept of "real vector" in unitary space barely merits the attention the authors pay it. The authors leave the reader with the impression that separability is an inherent rather than an 'assumed (and somewhat inessential) property of Hilbert space. At several points, the authors actually go counter to their purpose of developing the finite dimensional theory in a manner most closely adaptable to Hilbert space, and nothing is saved by their procedure ...