# Locally Convex Spaces and Linear Partial Differential Equations

• François Treves
Book

Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 146)

1. Front Matter
Pages I-XII
2. ### The Spectrum of a Locally Convex Space

1. Front Matter
Pages 1-1
2. François Treves
Pages 3-14
3. François Treves
Pages 15-24
4. François Treves
Pages 25-36
5. François Treves
Pages 37-42
6. François Treves
Pages 43-54
3. ### Applications to Linear Partial Differential Equations

1. Front Matter
Pages 55-55
2. François Treves
Pages 57-71
3. François Treves
Pages 72-82
4. François Treves
Pages 83-91
5. François Treves
Pages 92-104
4. Back Matter
Pages 105-123

### Introduction

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 orphic 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 approx­ imability -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.

### Keywords

Differential Equations Differential operator Differentialgleichung Lokalkonvexer Raum Manifold Partial Differential Equations Partielle Differentialgleichung differential equation equation function partial differential equation theorem

#### Authors and affiliations

• François Treves
• 1
1. 1.Purdue UniversityLafayetteUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-87371-3
• Copyright Information Springer-Verlag Berlin Heidelberg 1967
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-87373-7
• Online ISBN 978-3-642-87371-3
• Series Print ISSN 0072-7830