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Banach-Hilbert Spaces,
Vector
Measures
and
Group
Representations
'rsoy· Wo
Ma
Banach-Hilbert Spaces,
Vector Measures and
Group Representations
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Tsoy-Wo Ma
Un1vers1ty of Western Australia
World Scientific
New Jersey• London
•
Smgapore
•
Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P
0
Box 128, Farrer Road, Singapore 912805
USA office:
Suite 1B, 1060 Main Street, River Edge, NJ 07661
UK office:
57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
BANACH-HILBERT SPACES, VECTOR MEASURES AND
GROUP REPRESENTATIONS
Enlarged Edition of Classical Analysis on Normed Spaces
Copyright© 2002 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof. may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN 981-238-038-8
Printed in Singapore
by
World Scientific Printers (S) Pte Ltd
Preface
This book provides an elementary introduction to classical analysis on
normed spaces with special attention to nonlinear topics such as fixed points,
calculus and ordinary differential equations. In this second edition, a new
approach to vector measures on 5-rings based on dominated convergence is
introduced. Matrix-representations of groups are also included because mean-
values of almost periodic functions behave similar to translation-invariant
integrals and are available in infinite dimensional Banach spaces. This book is
for beginners who want to get through the basic material as soon as
possible and then do their own research immediately. It assumes only general
knowledge in finite dimensional linear algebra, simple calculus, elementary
complex analysis and in the last part also elementary group theory. The treat-
ment is essentially self-contained except chapter 27 which may be skipped
without discontinuity. With sufficient details, even an undergraduate with
mathematical maturity should have no trouble to work through it alone.
Various chapters can be integrated into parts of a Master Degree Program
by course work organized by any regional university. Restricted to
IRn
rather
than normed spaces, selected chapters can be used for a course in advanced
calculus. We also hope that Engineers and Physicists would find this book
to be a handy reference in classical analysis especially our approach to vector
measures. High school teachers may b.e interested to enrich their programs
by including the generalization of triangles and tetrahedra as treated in our
chapter 4. Some special features are highlighted below.
Banach-Hilbert Spaces
• Sequences can be interpreted as samples taken per unit time. It seems to be
more intuitive to use them as description of topological properties.
• Simplicial Complexes are treated in details for potential school projects if
restricted to
IR
2 ,
IR
3 •
• Transition from
IRr
to finite dimensional spaces is analytic, §5-1.9,12.
• Explicit fo.rmula for no retraction is given in §5-2.4.
e
Infinite dimensional topological results are developed without homology.
• Higher derivatives in addition to first derivatives are represented by matrices.
• Higher Chain-Products Formulas §§10-6.2,3 are expressed more naturally in
polynomials rather than in symmetric multilinear maps.
e
Local solution interval to initial valued problem is independent of initial data.
• Dependence on initial conditions are in global setting. As an informal illus-
tration, suppose a commodity can last one year in a laboratory. Local theory
VI
Preface
says that its mass production should work for at least a few seconds but our
global theory ensures a period of at least 300 days.
• Tensor products of vectors and linear maps are defined separately but we
prove that they are consistent in §15-5.11.
• To the best of our knowledge, tensor products of operators on Hilbert spaces
§§15-7.9 to 16 are our contribution.
Vector Measures
• Existing instructors do not have to learn new trick in order to demonstrate
their leadership in helping a new generation of scientists to equip with better
tools in
vector
measures-integrals.
• Our proposed approach can replace most existing courses in scalar
measure theory because the treatment is self-contained without assuming
Egorov's Theorem or semivariations from scalar-theory.
• Complex vector lattices are used as framework for measures and means.
• Breakable vector lattices ensure that order bounded linear forms are linear
combinations of positive linear forms .. This unifies several proofs in §§17-3.4,
24-6.9, 27-3.2, 31-3.2. · Some results of real vector lattices are extended to
complex breakable vector lattices.
• Semirings are the starting points of
all
our measures. Finite variation is
characterized in terms of order §17-3.6 and absolute convergence, §17-4.5.
• Measures are defined on 5-rings so that they need not be bounded. Sets in
5-rings are called
decent sets.
They correspond to bounded Borel sets in
IRn.
• Measures are of finite variation in order to use breakable vector lattices.
• Functions of finite variation are related to Stieltjes measures.
• Simple proof of certain complex charge to be of finite variation is given in
§18-1.4.
• Restriction to finite-valued outer measures allows the approximation of
eA.-tension to decent sets by values on sets in semirings, §18-3.3,4,6. As a result,
integrals of decent functions are defined.
• A set with a 5-ring is called a 5-space. Measurable sets are defined by
localization and are independent of all measures.
• Measurable functions in general are finite-valued and are defined everywhere
but J.t-measurable functions depend heavily on a particular measure J.t, §22-3.4.
• Approximation by simple functions has the additional property of increasing
modulus, §19-4.4.
• Explanation §19-5.1,3 why measurable (vector) maps should not be defined
trivially as sequential limits of simple maps as in most literatures so that
continuous maps are measurable.

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