Metric Geometry in Normed Spaces
Banach’s book, p. 160, gives a theorem of Mzaur and Ulam that an isometry of one normed space onto another which carries 0 to 0 is linear. This is true only for real-linear spaces, and is proved by characterizing the midpoint of a segment in a normed spac
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HERAUSGEGEBEN VON
L. V.AHLFORS · R. BAER · R. COURANT· J. L. DOOB · S. ElLENBERG P.R. HALMOS · M. KNESER·T. NAKAYAMA· H. RADEMACHER F.K. SCHMIDT· B. SEGRE ·E. SPERNER
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FOLGE ·HEFT 21
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REI HE:
REELLE FUNKTIONEN BESORGT VON
P.R.HALMOS
SPRINGER-VERLAG BERLIN HEIDELBERG GMBH 1958
NORMED LINEAR SPACES BY
MAHLON M. DAY
SPRINGER-VERLAG BERUN HEIDELBERG GMBH 1958
ALLE RECHTE, INSBESONDERE DAS DER UBERSETZUNG IN FREMDE SPRACHEN, VORBEHALTEN OHNE AUSDRUCKLICHE GENEHMIGUNG DES VERLAGES IST ES AUCH NICHT GESTATTET, DIESES BUCH ODER TEILE DARAUS AUF PHOTOMECHANISCHEM WEGE (PHOTOKOPIE, MIKROKOPIE) ZU VERVIELFALTIGEN
© BY SPRINGER-VERLAG BERLIN HEIDELBERG 1958 URSPRÜNGLICH ERSCHIENEN BEI SPRINGER-VERLAG BERLIN· GOTIINGEN ·HEIDELBERG 1958
ISBN 978-3-662-23231-6 ISBN 978-3-662-25249-9 (eBook) DOI 10.1007/978-3-662-25249-9
BRUHLSCHE UNIVERSITATSDRUCKEREI GIESSEN
Foreword 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. Urbana, Illinois (USA), September 1957
MARLON M. DAY
Contents
page
Chapter I. Linear spaces . . . . . . § 1. Linear spaces and linear dependence . § 2. Linear functions and conjugate spaces § 3. The Hahn-Banach extension theorem. § 4. Linear topological spaces . . § 5. Conjugate spaces . . . . . . § 6. Cones, wedges, order relations Chapter II. Normed Linear spaces . § 1. Elementary definitions and properties § 2. Examples of normed spaces; constructions of new spaces from old § 3. Category proofs . . . . . . . . . . . . § 4. Geometry and approximation . . . . . . . . § 5. Comparison of topologies in a normed space . . Chapter II I. Completeness, compactness, and reflexivity § 1. Completeness in a linear topological space . § 2. Compactness . . . . . . . . . . . § 3. Completely continuous linear operators . § 4. Reflexivity . . . . . . . . . . . . .
1 1 4 8 11 17 20 24 24 28 33 38 39 44 44 47 53 56
Chapter IV. Unconditional convergence and bases § 1. Series and unconditional convergence . . § 2. Tensor products of locally convex spaces § 3. Schauder bases in sepa
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