*Spectral Theory*(1962) and

*Precalculus: Fundamentals of Mathematical Analysis*(1973).

**1. Spectral Theory (1962), by E R Lorch. **

**1.1. From the Preface.**

... in the chapters on Banach spaces and linear transformation theory one will find the Hahn-Banach theorem, the inverse boundedness theorem, and the uniform boundedness principle; also the standard material on reflexivity, adjoint transformations, projections, reducibility, and even a formulation of the mean-ergodic theorem. The chapter on Hilbert space presents all the classic facts on linear functionals and orthonormal sets as well as the preliminary theory of self-adjoint transformations (bounded or not) and resolutions of the identity. Chapter IV is devoted to the Cauchy theory for operators. It contains the central facts of spectral theory: spectrum, resolvent, the fundamental projections, spectral radius, and the operational calculus. This theory is then applied to the problem of determining the structure of an arbitrary self-adjoint transformation in Hilbert space. Finally, in chapter VI, we consider Banach algebras. These are exclusively commutative and have a unit. We find here a discussion of reducibility, normed fields, ideals, residue rings, homomorphisms, and maximal ideals, the radical, the structure space, and the representation theory.

**1.2. Review by: B R Gelbaum.**

*Amer. Math. Monthly* **70** (3) (1963), 350.

This book is a translation of lectures delivered by the author in 1953-54 at the University of Rome where he was a Fulbright Visiting Professor. It is a model of economy and clarity. The reader is first guided through the essentials of Banach space lore, linear transformation theory and the specialization to Hilbert spaces. Then to provide an approach, different from that of most texts, to the spectral theorem, the author treats the integration of vector- (and operator-) valued functions over the complex plane. ... The book is highly recommended as an introduction to an interesting field, although detailed and far-reaching ramifications are untouched and a great deal of motivation is ignored-no doubt to save space. The author's use of language, choice of emphasis, organization of material and the selection of fringe benefits, (e.g., the mean ergodic theorem) should find a warm reception in a considerable mathematical audience.

**1.3. Review by: Kenneth Hoffman.**

*Science, New Series* **138** (3537) (1962), 132.

This is a valuable addition to the family of books on algebraic analysis. It provides an introduction to the theory of Banach spaces and Banach algebras, for the advanced undergraduate or young graduate student of mathematics and, perhaps, for some students of engineering or physics. Contrasted with recent tomes on the subject, it presents, with careful attention to detail, only the most basic results of the theory, in Lorch's inimitable style, which precludes the boredom common to many brief axiomatic developments. The material is treated through the spectral theory of linear operators, rather than by the other popular approach in which Banach algebras are made the basic vehicle. ... I would like to have seen a more detailed discussion of the examples from which the theory evolved. Also, it would have been nice to have this concise introduction to algebraic analysis include other basic tools of the discipline, such as the Stone-Weierstrass theorem and the Krein-Milman theorem. But, perhaps their inclusion would have sacrificed the brevity that is the book's most valuable asset.

**1.4. Review by: Louis de Branges.**

*Mathematical reviews* **MR0136967** (25 #427).

This text is an elementary exposition of linear transformations in Banach spaces suitable for graduate students with some preliminary knowledge of topology and integration. Proofs are given for the Hahn-Banach theorem, the uniform boundedness principle, the weak compactness of strongly closed and bounded sets of functionals, and the boundedness of everywhere defined, closed transformations. Hilbert spaces are introduced only in Chapter III, but prerequisites from earlier chapters are minimal. Due place is given to unbounded transformations in Hilbert space, though emphasis is on self-adjoint ones. Integration of projections yields examples of self-adjoint and unitary transformations. Chapter IV is devoted entirely to resolvents of transformations and the manipulation of related Cauchy integrals. The integral representation of self-adjoint transformations is obtained in Chapter V from general considerations applicable to transformations whose spectrum is real and whose resolvent is not too large near the real axis. Chapter VI expounds the Gel_fand representation of semi-simple commutative Banach algebras. Almost no examples and applications appear in this otherwise lucid introductory text.

**1.5. Review by: P M Anselone.**

*SIAM Review* **6** (3) (1964), 322.

This brief introduction to spectral theory is beautifully written. Ideas are carefully presented and elucidated. Demands on the reader are kept to a minimum: a little point set and metric topology, algebra, complex variables and integration theory. Nevertheless, topics of current interest are reached in a relatively few pages. The author accomplishes this by rejecting "encyclopaedic tendencies." In his words, "This is a kernel and not a hull presentation." ... Unfortunately, only a few examples and problems are given. Another omission is a treatment of compact operators. It would seem advisable to supplement this book with appropriate material in order to use it as a beginning textbook. Incidentally, one possible supplement might be the author's article, "The Spectral Theorem," in "Studies in Modern Analysis," edited by R C Buck, MAA & Prentice-Hall, 1962.

**2. Precalculus: Fundamentals of Mathematical Analysis (1973), by E R Lorch.**

**2.1. Review by: Norman Schaumberger.**

*The Mathematics Teacher*

**66**(7) (1973), 638.

A flexible introduction to precalculus mathematics that succeeds in making the subject more logical without increasing its difficulty. The instructor who emphasizes applications will like the wealth of exercises that range from drill problems to those that challenge the better student. All of the standard material is included, and the instructor who stresses theory will find a clear, interesting, and honest exposition. Interspersed among the customary topics are sections on maximum and minimum, famous inequalities, special methods of graphing, square roots and nth roots, the behaviour of polynomials at infinity, the unique factorization theorem, and numerous historical notes. The text has a mathematical spark that should make teaching it a pleasure.