Minimal Projections in Banach Spaces Problems of Existence and Uniqu
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Wlodzimierz Odyniec Pedagogical University Nab reki Mojki 48 Leningrad 191186, USSR Grzegorz Lewicki Jagiellonian University, Department of Mathematics Reymona 4, 30-059 Krakow, Poland
Mathematics Subject Classification (1980): 41A35, 41A52, 41A65, 46B99, 47A30, 47B05,47B38 ISBN 3-540-53197-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-53197-1 Springer-Verlag New York Berlin Heidelberg
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© Springer-Yerlag Berlin Heidelberg 1990 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper
To Marii Schiffmann CElenbogen)
Preface
This book
in
from
Pedagogical
held by
of Bydgoszcz in 1984 - 85.
A
of
I. II. IV) is a revised version of [156J. which appeared in 1985 under copies. in
same
form of mimeographed
(in Russian) in 350
(a
issue).
III includes new The
is
an
Index and
chapter is followed by short
Index; each
on
of
exposed
in the chapter. It is
hope that
book may be of
are just beginining to deepen and approximation
to
wh(
knowledge of Banach space
as well as to specialists. The second chapter
can perhaps prove to be
also
specialists in
economy. This book would never have appeared
encouraging help from
many persons. First of all. the his suggestion
are grateful
Professor Ward Cheney for
submit
and
series and for his inte-
along
Also our
go
Professors Yu.
Konjagin. A. Pelczynski. S. A. Zalgaller. Dr.
way
V. M.
discussions
D. Burago. M.
Rolewicz. P.
Kadec. Dr.
is due
the
version of
Kadec. S.
A. W.
A. Koldobskii and Dr.
problems dealt with in
Special
I. W.
K.
for
book.
Professor Czeslaw Bessaga who has read and helped a
in
final version. Finally. we are glad cikiewicz and Miss J. The
thank Dr.
be able
M.
W6jcikiewicz who
also would like
ran for his help in preparing
E.
Kuczma. Mr M.
W6j'
English.
Dr.
express final version of
M.
Ba-
book on computer.
Wlodzimierz Odyniec and Grzegorz Lewicki Krak6w. April 1990
Table of Contents
1
Introduction §
1. General survey
1
§
2. Terminology and notation
14
Chapter I. Problem of uniqueness of minimal projections
18
in Banach Spaces §
1. Uniqueness of minimal projection with nonunit norm
18
in a three-dimensional Banach space §
2. Certain properties of minimal projections
24
§
3. On the uniqueness or nonuniqueness of minimal projections
32
in the case of dim (B/D) §
2
39
4. Strong uniqueness of minimal projections Notes and remarks
50
Chapter II.
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