We introduce and study a new class of locally convex vector lattices of continuous functions on a locally compact Hausdorff space, which we call regular vector lattices. We investigate some general properties of these spaces and of the subspaces of so-called generalized affine functions. Moreover, we present some Korovkin-type theorems for continuous positive linear operators; in particular, we study Korovkin subspaces for finitely defined operators, for the identity operator and for positive projections. Due to its length, the paper is split up into two parts of distinct character; in this first part, we introduce the class of regular vector lattices, we prove an integral representation theorem for continuous positive linear forms and we study some enveloping functions related to a continuous positive operator, together with the corresponding space of generalized affine functions. Finally, we obtain a Stone-Weierstrass type theorem. In the second part, which will appear in the same journal, we will present some Korovkin-type theorems, together with some applications.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-3-3,
author = {Francesco Altomare and Mirella Cappelletti Montano},
title = {Regular vector lattices of continuous functions and Korovkin-type theorems-Part I},
journal = {Studia Mathematica},
volume = {166},
year = {2005},
pages = {239-260},
zbl = {1092.46018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-3-3}
}
Francesco Altomare; Mirella Cappelletti Montano. Regular vector lattices of continuous functions and Korovkin-type theorems-Part I. Studia Mathematica, Tome 166 (2005) pp. 239-260. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-3-3/