L. V. Ahlfors, and Others, Analytic Functions (Princeton University Press, 1960), vii+197 pp., 40s.
D. Martin · Proceedings of the Edinburgh Mathematical Society · 1962
In Chapter IX, real Hilbert space is defined axiomatically, and after some discussion of subspaces and of bilinear functionals, the concept of a self-adjoint operator is introduced.The representation of a completely continuous self-adjoint operator in terms of its eigenvalues and eigenvectors is established, and, finally, the usefulness of this is illustrated by a brief consideration of integral equations with symmetric L 2 kernels.It seems odd that there is no mention of complex Hilbert space (even though there is a reference, at the end of Chapter VIII, to quantum theory); but as with Volume 1, it is surprising to find so much material in so few pages, without undue compression.The authors have again achieved a nice blend of the abstract and the concrete.The Graylock translation benefits from the inclusion of a number of exercises, prepared by H. Kamel, which are distributed through the book and usefully amplify the text.An appendix, translated from the Russian edition of this volume, corrects some minor errors in Volume 1.Here, in the Graylock translation, is a very good book: an excellent companion to the first volume, and also to Kolmogorov's well-known little book on probability (of which an up-to-date English edition would now be welcome).