Francisco Montalvo

Uniform approximation theorems for real-valued continuous functions

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) ¹ 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-Molto for the bounded case [2].

Order topologies on l-algebras

We consider different topologies on an l-algebra associated in a natural way with its order, and briefly study the relationship between them. As a consequence we obtain a partial answer to the problem of the characterization of the topological algebra C(X) of real continuous functions on a topological space X, endowed with its compact convergence topology. In fact, we prove that if A is an Archimedean l-algebra and its order topology τo is locally m-convex, bornological and complete, then there is a realcompact kr -space X such that (A, τo) is isomorphic and homeomorphic with C(X).  2003 Elsevier B.V. All rights reserved. MSC: 06B30; 54H12; 46H05

Algebraic Properties of the Uniform Closure of Spaces of Continuous Functions

For a completely regular space X, C(X) denotes the algebra of all real‐valued and continuous functions over X. This paper deals with the problem of knowing when the uniform closure of certain subsets of C(X) has certain algebraic properties. In this context we give an internal condition, “property A,” to characterize the linear subspaces whose uniform closure is an inverse‐closed subring of C(X).

Generation of the uniformly continuous functions

Let X be a set and F a family of real-valued functions on X. We denote by μFX the space X endowed with the weak uniformity given by F. In this paper we provide a method of generating the set U(μFX), of all uniformly continuous real functions on μFX, by means of the family F. In order to do that we need to study the uniform approximation of real uniformly continuous functions on subsets of Rn. As a consequence, we give an internal condition on F in order to be uniformly dense in U(μFX).

Some density results for uniformly continuous functions

Abstract Let X be a set and F a family of real-valued functions (not necessarily bounded) on X . We denote by μ F X the space X endowed with the weak uniformity generated by F. and by U ( μ F X ) the collection of uniformly continuous functions to the real line R. In this note we study necessarily and sufficient conditions in order that the family F, be uniformly dense in U ( μF X ). Firstly, we give a ore direct proof of a result by Hager involving an external condition over F given in terms of composition with the uniformly continuous and real-valued functions defined on subsets of R n . From this external condition we can derive as easy corollaries most of the results already known in this context. In the second part of this note we obtain an internal necessary and sufficient condition of uniform density set by means of certain covers of X by cozero-sets of functions in F.