Differential Calculus in Topological Linear Spaces
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374 Sadayuki Yamamuro
Differential Calculus in Topological Linear Spaces
Springer-Verlag Berlin. Heidelberg. New York 1 974
AMS Subject Classifications (1970): 46-02, 46 B99, 4 6 G 0 5 , 58-02, 58 C 20 ISBN 3-540-06709-4 Springer-Verlag Berlin - Heidelberg • New York ISBN 0-387-06709-4 Springer-Verlag New York • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1974. Library of Congress Catalog Card Number 73-21376. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
CONTENTS
1
Introduction Chapter i.
Definitions and fundamental properties
6
§i.i
Directional derivatives
6
§1.2
M-derivatives
7
51.3
Mean value theorems
13
§1.4
Relations among
17
§i.5
Differentiability in spaces with projective topology
19
§1.6
Differentiability in inductive limits of increasing families of subspaces.
21
§t.7
Differentiability and continuity
23
§1.8
Higher derivatives
25
§1.9
Equicontinuously differentiable mappings
30
§1.10
Uniform differentiability
34
§1.11
Partial derivatives
36
§1.12
Other differentiabilities
39
C h a p t e r 2.
M-differentiabilities
43
Compact mappings
§2.1
Compact mappings and Fr~chet derivatives
43
§2.2
Compact mappings and Hadamard difTerentiability
47
Chapter 3.
Inverse mapping theorems
50
§3.1
Differentiation in
§3.2
Differentiability of inverse mappings
53
53.3
The space
58
§3.4
C -mappings and an inverse mapping theorem P Other theorems on inverse mappings
§3.5
L(E,F)
5O
Lp(E,F)
61 69
3~ Chapter 4.
76
Differentiability of semi-norms
§4.1
Hadamard differentiability of semi-norms
76
§4.2
Frechet differentiability of semi-norms
81
§4.3
Higher derivatives of semi-norms
84
§4.4
Differentiability of the supremum norms of function spaces
87
§4.5
Differentiability of norms of
95
Chapter 5.
Lp-spaces
Smoothness
98 98
§5.1
S-categories
§5.2
S-smooth spaces
I00
§5.3
Partitions of unity
106
Chapter 6.
Differentiability of mappings of a real variable
iii
§6.1
Differentiability of Lipschitz mappings
iii
§6.2
Differentiability of Stepanoff mappings
114
§6.3
Theorems of L. Schwartz and A. Grothendieck
121
Chapter 7.
Sets of differentiable mappings
124
§7.1
Idempotents of semigroups of differentiable mappings
124
§7.2
Automorphisms of semigroups of differentiable mappings
126
§7.3
Near-rings of differentiable mappings
132
Appendix i.
Sequential spaces
140
Appendix 2.
Continuity of composition mappings
143
Appendix 3.
Differentiability of inverse mappings
147
List of symbols
155
References
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