The chief aim of this book, which we trust is not unrealistic, is to accustom advanced students of mathematics, through systematically arranged problems in some important fields in analysis, to the ways and means of independent thought and research. It is intended to serve the need for individual active study on the part of both the student and the teacher. The book may be used by the student to extend his own reading or lecture material, or he may work quite independently through selected portions of the book in detail. The instructor may use it an an aid in organising tutorials or seminars.
This book is no mere collection of problems. Its most important feature is the systematic arrangement of the material which aims to stimulate the reader to independent work and to suggest to him useful lines of thought. We have devoted more time, care and detailed effort to devising the most effective presentation of the material than might be apparent to the uninitiated at first glance.
The imparting of factual knowledge is for us a secondary consideration. Above all we aim to promote in the reader a correct attitude, a certain discipline of thought, which would appear to be of even more essential importance in mathematics than in other scientific disciplines.
One should try to understand everything: isolated facts by collating them with related fact, the newly discovered through its connection with the already assimilated, the unfamiliar by analogy with the accustomed, special results through generalisation, general results by means of suitable specialization, complex situations by dissecting them into their constituent parts, and details by comprehending them within a total picture.
One must not forget that there are two kinds of generalisation, one facile and one valuable. One is generalisation by dilution, the other is generalization by concentration. Dilution means boiling the meat in a large quantity of water into thin soup, concentration means condensing a large amount of nutritive material into an essence. The unification of concepts which in the usual view appear to lie far removed from each other is concentration. Thus, for example, group theory has concentrated ideas which formerly were found scattered in algebra, number theory, geometry and analysis and which appeared to be very different. Examples of generalisation by dilution would be still easier to quote, but this would be at the risk of offending sensibilities.