Matrix Methods in Analysis
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1113 Piotr Antosik Charles Swartz
Matrix Methods in Analysis
Springer-Verlag Berlin Heidelberg New York Tokyo 1985
Authors
Piotr Antosik Institute of Mathematics, Polish Academy of Sciences ul. Wieczorka 8, 40-013 Katowice, Poland Charles Swartz Department of Mathematical Sciences, New Mexico State University Las Cruces, N.M. 88003, USA
AMS Subject Classification (1980): Primary: 46-0, Secondary: 28C20, 40A05, 40C05, 46A15, 46G10 ISBN 3-540-15185-0 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15185-0 Springer-Verlag New York Heidelberg Berlin Tokyo 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 "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Contents 1.
Introduction......................
1
Introductory remarks, outline, notation and terminology, Drewnowski Lemma. 2.
Basic Matrix Results.
7
Basic Matrix Theorem. 3.
X -convergence
.....•................
Definition of X-convergence, examples, X-boundedness,
X-space,
X-convergence doesn't imply
X-convergence and
11
X-bounded,
X-boundedness in the weak and weak*
topologies. 4.
The Uniform Boundedness Principle
. . . . . . . . .
21
General Uniform Boundedness Principle, Classical Uniform Boundedness Principle, Nikodym Boundedness Theorem.
S.
Convergence of Operators . . . . . . . . . . . . . .
29
Classical Banach-Steinhaus Theorems, General BanachSteinhaus Theorem, the Brooks-Jewett Theorem, Nikodym Convergence and Vitali-Hahn-Saks Theorems, the HahnSchur
6.
Theorem.
Bilinear Maps and Hypocontinuity X-hypocontinuity, separate continuity implies joint continuity, separate continuity and equicontinuity for sequences of bilinear maps, separate equicontinuity and equihypocontinuity.
.
4S
IV
7.
Orlicz-Pettis Theorems . . . . .
58
Classical Orlicz-Pettis Theorem, Tweddle-type OrliczPettis Theorem, Orlicz-Pettis result for compact operators Orlicz-Pettis Theorems for the topology of pointwise convergence in function spaces, abstract Orlicz-Pettis Theorems.
8.
The Schur and Phillips Lemmas
......••
74
General Schur Lemma, Classical Schur Lemma, General Hahn-Schur Summability Result, General Phillips' Lemma, Classical Phillips' Lemma. 9.
The Schur Lemma for Bounded Multiplier Convergent Series. .
85
General Schur Lemma for Bounded Multiplier Convergent Series, Summability Result, C-convergent series, Schur Lemma for C-convergent series. 10.
Imbedding Co and t""
. . . . . . . . . . . . . • . .
94
Weak unconditional Cauchy series, Bessaga-Pelczynski Theorem, Diestel's Theorems, Pelczynski's Theorem on unconditionally converging operators, Diest
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