Introduction to Linear Algebra, by Frank M. Stewart. D. van
Nostrand Co. , Inc., Princeton, 1963. xv + 281 pages.
This is a carefully written, well thought out book which can be
used successfully for either an elementary, introductory course in linear
algebra for students with no background in the subject or for a more
sophisticated course for students who may already be familiar with some
of the computational aspects of matrix theory. Features of this book
include: the determinant is developed in terms of multilinear forms
and then, in a completely separate and independent chapter, is redeveloped
using the classical approach; appendices on basic logic, sets, proofs,
functions, etc. are provided and are referred to throughout the text by
marginal footnotes; there is an index of symbols as well as a general
index and a table giving the page number for each theorem, definition
and corollary.
M. Pearl, University of Maryland
Introduction to Topological Groups, by Taqdir Husain.
W.B.
Saunders Co., Philadelphia and London, 1966. xi + 218 pages.
£8.10.
This is an introductory text on topological groups. Except for
Chapters II and V, much of the material is standard. The semitopological
groups studied in Chapter II are groups endowed with a topology so that
only the multiplication is continuous in each variable separately. One
finds here several conditions for a semitopological group to be a topo-
logical group. Chapter V is an unusually thorough discussion on open
homomorphisms and closed graph theorems.
The existence and essential uniqueness of the Haar integral are
proved in Chapter VI. Representations of compact groups are studied
in Chapter VII, which includes a section on integral equations used in
the proof of Peter-Weyl's theorem. In Chapter VIII, dual groups of
locally compact abelian groups are introduced; Bochner-Weil's theorem
on positive definite functions and the Plancher el theorem are proved.
The last chapter is an introduction to Banach algebras.
As the book is intended to be an introduction to the theory of
topological groups, several important topics (e.g., Gelfand-Raikov's
theorem on unitary representations of locally compact groups,
Pontryagin's duality theorem, the structure theorem for locally compact,
compactly generated abelian groups) are not included. The book is
clearly written and suitable for a beginning course on topological groups.
Ky Fan, University of California, Santa Barbara
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