Prove that closed unit ball of L^1 (relative to lebesgue measure on the unit int

Question

Prove that closed unit ball of L^1 (relative to lebesgue measure on the unit interval) has no extreme points but every point on the surface of the unit ball in L^p (1<p<infity) is an extreme point of the ball.

Explanation / Answer

00 OU 160181 >E =; oo OSMANIA UNIVERSITY LIBRARY Call No. 5 2> V^H G^b Accession No. 4- 89 *XO Author Ho{..vr*x-r KeirvTvV - This lx>ok should be returned on or before the date last marked below. BANACH SPACES of ANALYTIC FUNCTIONS PRENTICE-HALL SERIES IN MODERN ANALYSIS R. CREIGHTON BUCK, editor BANACH SPACES ANALYTIC FUNCTIONS KENNETH HOFFMAN Department oj Mathematics Massachusetts Institute of Technology PRENTICE-HALL, INC. Englewood Cliffs, N. J., 1962 1962 by Prentice-Hail, Inc., Englewoocl Cliffs, N. J. A II rights reserved. No part of this book may be repro- duced in any form, by mimeograph or any other means, without permission in uniting from the publisher. Library of Congress Catalog Card Number: 62-12452 PRINTED IN THE UNITED STATES OF AMERICA 05540 C to Pat PREFACE There are not enough books which deal with the interplay between func- tional analysis and the theory of analytic functions. One reason for this is the fact that many of the techniques of functional analysis have a "real variable" character and are not directly applicable to problems which belong intrinsically to analytic function theory, e.g., problems of conformal mapping and Riemann surfaces. But there are parts of this theory which blend beautifully with the concepts and methods of functional analysis. These are fascinating areas of study for the general analyst, for three prin- cipal reasons: (a) the point of view of the algebraic analyst leads to the formulation of many interesting problems concerned with analytic func- tions; (b) when such problems are solved by a combination of the tools from the two disciplines, the depth of each discipline is increased; (c) the techniques of functional analysis often lend clarity and elegance to the proofs of classical theorems, and thereby make the results available in more general situations. The main purpose of this monograph is to provide an introduction to the segment of mathematics in which functional analysis and analytic function theory merge successfully. Its spirit is close to that of abstract harmonic analysis, and, in fact, there is some overlap with the subject matter of harmonic analysis. Because this work is introductory, there has been no attempt to emulate cither the depth of Zygmund's book on trigo- nometric series or the generality of the several books which treat harmonic analysis on groups. The subject matter is restricted to Banach spaces of analytic functions in the unit disc, roughly, those which are closely related to the Hardy spaces H p (1 g p ^ oo). The historical accounting some- times falls a bit short of the mark. Some effort toward such an accounting is made in the sections entitled NOTES, at the end of each chapter. But a few relevant references have been omitted (for example, A. Taylor's papers in Studio, Mathematica, 1950-51). The material is not discussed in its ulti- mate generality. Where proofs do carry over to more general contexts and the extension is not treated elsewhere, my method is usually to give the proofs in the unit disc and to discuss the generalizations afterward. The first four chapters are devoted to the proofs of classical theorems on boundary-values and boundary integral representations for analytic vii viii Preface functions in the unit disc which lie in the Hardy class H p (1 ^ p ^

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