Schauder Bases in Banach Spaces of Continuous Functions
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Zbigniew Semadeni
Schauder Bases in Banach Spaces of Continuous Functions
Springer-Verlag Berlin Heidelberg New York 1982
Author
Zbigniew Semadeni lnstytut Matematyczny, Polskiej Akademii Nauk ul. Sniadeckich 8, skr. pocztowa 137,00·950 Warszawa
AMS Subject Classifications (1980): 41 A 15,46 BXX, 46B15, 46B25, 46B30, 46E15
ISBN 3-540-11481-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11481-5 Springer-Verlag New York Heidelberg Berlin
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© by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Preface Writing these lecture notes I had several goals in mind: 1)
To present (in Chapters 1 and 4)
Schauder bases in spaces
C(X) and
theoretical aspects of
Co(X),
in the spirit of the iso-
metric theory of Banach spaces, including a construction of a basis in a generic separable space
Co(X).
2) To show (in Chapter 2) constructions of some classical bases of functions of a single variable. This includes the FaberSchauder system in
C(I),
where
r = [0,1],
and the Haar system as a basis in
the Franklin system in
C(I),
C(2 w) .
3) To give an introduction to the work of Z.Ciesielski, S.Schoenefeld and others on spline bases in various spaces of differentiable functions and to the work of
on bases in certain Banach
spaces of analytic functions (Section 3.9). 4) To present (in Section 4.8) results concerning nonexistence of
Schauder
bases
with
some
special properties; typical here is
the celebrated theorem of A.M.Olevskii that no uniformly bounded orthonormal system can be a basis
in
C(I).
5) To show (in Chapter 3) constructions of bases in spaces where
1 < d < co.
C(I
d)
For two (or perhaps even four) decades it has been
known how to construct such bases, consisting of certain spline functions; in these notes we consider bases of splines of degree 1, d that is, of functions continuous on r and affine on each simplex of some triangulation of
rd.
In spite of the regularity of the construc-
tion of these bases and their nice properties, they have not yet attracted people working in numerical methods. An
reason is
that not only these functions are not differentiable, but also the rate of convergence is worse than with higherorder splines. Yet, another reason may be that in the existing literature the descriptions of the basis functions of several variables are geometrical, outlined only, without explicit formulas for the functions. This is why the d) presentation of bases in C(I is so detailed here. Also, though arra.nging the basis functions into a single sequence is
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