Absolute Analysis
The first edition of this book, published in German, came into being as the result of lectures which the authors held over a period of several years since 1953 at the Universities of Helsinki and Zurich. The Introduction, which follows, provides informati
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Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berucksichtigung der Anwendungsgebiete Band 102
fferausgegehen von
J. L. Doob
A. Grothendieck E . Heinz F. Hirzebruch E. Hopf W. Maak S. MacLane W. Magnus J. K. Moser M. M. Postnikov F. K. Schmidt D. S. Scott K. Stein
C;escba~t~uhrende
ff erausgeher
B. Eckmann und B. L. van der Waerden
F. and R. Nevanlinna
Absolute Analysis Translated from the German
by Phillip Emig
With 5 Figures
Springer-Verlag Berlin Heidelberg GmbH
F. and R. Nevanlinna Department of Mathematics, University, Helsinki/finland Translat or
P . Emig Granada Hills. CA 91344/USA
Gesch!iJtsiQhrende Herausgeber
B. Eckmann £idgenOssische Tcchnische Hochschule Zurich
B. L. van der Waerden Mathematisches I nstitut der Universitat Zlirich
AMS Subject Classifications (1970): 26 A 60
ISBN 978-3-662-00251-3 ISBN 978-3-662-00249-0 (eBook) DOI 10.1007/978-3-662-00249-0
This work is subjekt copyright. All rights are reserved, wheter the whole or pan of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amonU I of the fee to be determined by agreement with the publisher. 0 by Springer-Verlag Berlin Heidelberg 1973. Originally published by Springer-Verlag Berlin Heidelberg New York in 1973. Softcover reprint of the hardcover 1st edition 1973 . Library of Congress Catalog Card Number 73-75652.
Foreword The first edition of this book, published in German, came into being as the result of lectures which the authors held over a period of several years since 1953 at the Universities of Helsinki and Zurich. The Introduction, which follows, provides information on what motivated our presentation of an absolute, coordinate- and dimension-free infinitesimal calculus. Little previous knowledge is presumed of the reader. It can be recommended to students familiar with the usual structure, based on coordinates, of the elements of analytic geometry, differential and integral calculus and of the theory of differential equations. We are indebted to H . Keller, T . Klemola, T. Nieminen, Ph. Tondeur and K. 1. Virtanen, who read our presentation in our first manuscript, for important critical remarks. The present new English edition deviates at several points from the first edition (d. Introduction). Professor I. S. Louhivaara has from the beginning to the end taken part in the production of the new edition and has advanced our work by suggestions on both content and form. For his important support we wish to express our hearty thanks. We are indebted also to W. Greub and to H. Haahti for various valuable remarks. Our manuscript for this new edition has been translated into English by Doctor P. Emig. We express to him our gratitude for his careful interest and skillful attention during this work. Our thank
