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On Schauder Bases for Spaces of Continuous Functions1)

Published online by Cambridge University Press:  20 November 2018

H. W. Ellis  and
D. G. Kuehner
D. G. Kuehner
Affiliation:
Summer Research Institute and Queen’s University
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In a finite dimensional vector space V a set xi, i = 1, 2, …, n of vectors of V is said to be a basis, base, or coordinate system for V if the vectors xi are linearly independent and if each vector in V is a linear combination of the elements x1 with real coefficients. If a topology for V is defined in terms of a norm ||.|| then {xi} is a basis for V if and only if to each x ϵ V corresponds a unique set of constants ai such that

In infinite dimensional normed vector spaces the above concepts of basis have different generalizations. The first or algebraic definition gives a Hamel basis which is a maximal linearly independent set [l, p. 2]. We shall be interested in the other or topological definition.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 3 , Issue 2 , 01 May 1960 , pp. 173 - 184
Copyright
Copyright © Canadian Mathematical Society 1960

Footnotes

2)

Holder of a National Research Council Studentship.

1)

Part of this note is adapted from Mr. Kuehner's M.A. thesis.

References

1. Day, M. M., Normed Linear Spaces (Berlin, 1958).Google Scholar
2. Ellis, H. W. and Israel, Halperin, Haar functions and the basis problem for Banach spaces, J. London Math. Soc, 31 (1956), 28-39.Google Scholar
3. Kelley, J. L., General Topology (New York, 1955).Google Scholar
4. Loomis, L. H., An Introduction to Abstract Harmonic Analysis (New York, 1953).Google Scholar
5. Schauder, J., Zur Th?orie stetiger Abbildungen in Funktionalr?umen, Math. Z. 26 (1927), 47-65.Google Scholar
6. Zygmund, A., Trigonometrical Series (Warsaw, 1935).Google Scholar