For a completely regular space X, C(X) and C*(X) denote, respectively, the algebra of all real-valued continuous functions and bounded real-valued continuous functions over X. When X is not a pseudocompact space, i.e., if C*(X) ≠ C(X), theorems about uniform density for subsets of C*(X) are not directly translatable to C(X). In [1], Anderson gives a sufficient condition in order for certain rings of C(X) to be uniformly dense, but this condition is not necessary.
In this paper we study the uniform closure of a linear subspace of real-valued functions and we obtain, in particular, a necessary and sufficient condition of uniform density in C(X). These results generalize, for the unbounded case, those obtained by Blasco-Moltó for the bounded case [2].
@article{urn:eudml:doc:39941,
title = {Uniform approximation theorems for real-valued continuous functions.},
journal = {Extracta Mathematicae},
volume = {6},
year = {1991},
pages = {152-155},
mrnumber = {MR1185365},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:39941}
}
Garrido, M. Isabel; Montalvo, Francisco. Uniform approximation theorems for real-valued continuous functions.. Extracta Mathematicae, Tome 6 (1991) pp. 152-155. http://gdmltest.u-ga.fr/item/urn:eudml:doc:39941/