Locally Convex Spaces and Linear Partial Differential Equations
It is hardly an exaggeration to say that, if the study of general topolog ical vector spaces is justified at all, it is because of the needs of distribu tion and Linear PDE * theories (to which one may add the theory of convolution in spaces of hoi om o
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JSreraus~e~eben
von
J. L. Doob
. E. Heinz· F. Hirzebruch . E. Hopf . H. Hopf W. Maak . S. Mac Lane' W. Magnus· D. Mumford M. M. Postnikov . F. K. Schmidt· D. S. Scott· K. Stein
GeschiiftsJiihrende JSreraus~eber B. Eckmann und B. L. van der Waerden
Fran~ois
Treves
Locally Convex Spaces and Linear Partial Differential Equations
Springer-Verlag New York Inc. 1967
Professor Fran~ois Treves Purdue University, Lafayette, Indiana, USA. Geschiiftsfiihrende Herausgeber: Prof. Dr. B. Eckmann Eidgenossische Technische Hochschule ZUrich Prof. Dr. B. L. van der Waerden Mathematisches Institut der Universitiit ZUrich
ISBN- \3: 978-3-642-87373-7 DOl: 10.1007/978-3-642-87371-3
e-lSBN-13: 978-3-642-87371-3
All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard) or by other procedure without written permission from Springer-Verlag II:! by Springer-Verlag Berlin· Heidelberg 1967. Library of Congress Catalog Card Number 67-25286 Softcover reprint of the hardcover 1st edition 1967
Tide-No. 5129
To Ursula
Introduction It is hardly an exaggeration to say that, if the study of general topological vector spaces is justified at all, it is because of the needs of distribution and Linear PDE * theories (to which one may add the theory of convolution in spaces of hoi omorphic functions). The theorems based on TVS ** theory are generally of the "foundation" type: they will often be statements of equivalence between, say, the existence - or the approximability - of solutions to an equation Pu = v, and certain more "formal" properties of the differential operator P, for example that P be elliptic or hyperboJic, together with properties of the manifold X on which P is defined. The latter are generally geometric or topological, e.g. that X be P-convex (Definition 20.1). Also, naturally, suitable conditions will have to be imposed upon the data, the v's, and upon the stock of possible solutions u. The effect of such theorems is to subdivide the study of an equation like Pu = v into two quite different stages. In the first stage, we shall look for the relevant equivalences, and if none is already available in the literature, we shall try to establish them. The second stage will consist of checking if the "formal" or "geometric" conditions are satisfied. Each one of these phases requires specific techniques: checking of the formal or the geometrical conditions generally demands "hard analysis" methods, might for instance require the construction of a fundamental solution, or the proof of uniqueness in a Cauchy problem. The proof of the equivalences - the first step - will usually rely on "soft analysis", that is, on the study of rather poor structures, such as those of some brand of locally convex spaces. The present book is concerned with the soft analysis, applied to linear PDE's. It is essentially expository, and does not contain any new result on the subject of partial differ
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