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This book contains a compressed introduction to the study of normed linear spaces and to that part of the theory of linear topological spaces without which the main discussion could not well proceed. Definitions of many terms which are required in passing can be found in the alphabetical index, page 134. Symbols which are used throughout all, or a significant part, of this book are indexed on page 132. Each reference to the bibliography, page 124, is made by means of the author's name, supplemented when necessary by a number in square brackets. The bibliography does not completely cover the available literature, even the most recent; each paper in it is the subject of a specific reference at some point in the text. The writer takes this opportunity to express thanks to the University of Illinois, the National Science Foundation, and the University of Washington, each of which has contributed in some degree to the cultural, financial, or physical support of the writer, and to Mr. R. R. PHELPS, who eradicated many of the errors with which the manuscript was infested.
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