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.
Kuehner, D. G. On Schauder Bases for Spaces of Continuous Functions1). Canadian mathematical bulletin, Tome 3 (1960) no. 2, pp. 173-184. doi: 10.4153/CMB-1960-022-4
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author = {Kuehner, D. G.},
title = {On {Schauder} {Bases} for {Spaces} of {Continuous} {Functions1)}},
journal = {Canadian mathematical bulletin},
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year = {1960},
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number = {2},
doi = {10.4153/CMB-1960-022-4},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1960-022-4/}
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