Calculus in Vector Spaces without Norm
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A. Frolicher- W. Bucher Universite de Geneve
Calculus in Vector Spaces without Norm 1966
Springer-Verlag· Berlin· Heidelberg· New York
Work supported in part by the Swiss National Science Foundation
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, mlcrofilm and/or mlcrocard) or by other procedure without written permisslon from Springer Verlag. C by Springer-Verlag Berlin' Heidelberg 1966. Library of Congress Catalog Card Number 66-30345. Printed in Germany. Title No. 7350
- I -
CON TEN T S
Introduction §l. ELEMENTARY PROPERTIES Of fILTERS 1.1 1.2 1.3 1.4 1.5
filters and filter-basis Comparison of filters on the same space Mappings into direct products Images of filters under mappings An inequality between images of filters
§2. PSEUDO-TOPOLOGICAL VECTOR SPACES 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Pseudo-topological spaces Continuity Induced structures Pseudo-topological vector spaces Quasi-bounded and equable filters Equable pseudo-topological vector spaces The associated locally convex topological vector space Equable continuity Continuity with resrect to the structures
§3. DIFFERENTIABILITY AND DERIVATIVES 3.1 3.2 3.3 3.4
Remainders Differentiability at a point The chain rule The local caracter of the differentiability condition
§4. EXAMPLES AND SPECIAL CASES 4.1 4.2 4.3
The classical case Linear and bilinear maps The special case f: R-E
IV 1 1
2 2
:3 4
6
6
8 8
12 15 19 21 24 30 32 32 35 38 38 42 42 43
44
- II -
4.4 §5.
§6.
§7.
§9.
§1O.
46
FUNDAMENTAL THEOREM OF CALCULUS
50
5.1 5.2 5.3
50
formulation and proof of the main theorem Remarks and special cases Consequences of the fundamental theorem
58 60
PSEUDO-TOPOLOGIES ON SOME fUNCTION SPACES
65
6.1 6.2 6.3 6.4
65
The spaces B(E C and L(El;E2) l;E2), o(E l;E2) Continuity of evaluation maps Continuity of composition maps Some canonical isomorphisms
69
71 72
THE CLASS OF ADMISSIBLE VECTOR SPACES
82
7.1 7.2 7.3
82 84
7.4 §8.
Differentiable mappings into a direct product
The admissibility conditions Admissibility of EAdmissibility of subspaces, direct products and projective limits Admissibility of B(E C L l;E2), o(E l;E2), p(E l;E2)
85 87
PARTIAL DERIVATIVES AND DIFFERENTIABILITY
90
8.1 8.2
90 91
Partial derivatives A sufficient condition for (total) differentiability
HIGHER DERIVATIVES
93
9.1 9.2
93 95
and the symmetry of f"(x) f(P) for p !it 1
f"
Ck-fYIAPPINGS 10.1 The vector space 10.2 The structure of
99 Ck(E l;E2) Ck(El;E2)
10.3 C- (E1; E2 ) 10.4 Higher order chain rule
99 101 lOll 105
- III -
§11. THE compOSITION Of Ck-MAPPINGS
11.1 11.2
The continuity of the composition map The differentiability of the composition map
§12. DIFFERENTIABLE DEFORmATION OF DIFfERENTIABLE mAPPINGS 12.1 12.2
The differentiability of the evaluation map The linear homeomorphism
C: (£l;C:' «(2;E 3» - C:'(E l Ie E2;E3) APPENDIX NOTATIONS
INDEX REFERENCES
110 110 114 131 131
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