Regular vector lattices of continuous functions and Korovkin-type theorems—Part I

Francesco Altomare¹ ; Mirella Cappelletti Montano¹

¹ Department of Mathematics University of Bari Campus Universitario Via E. Orabona, 4 70125 Bari, Italy

Studia Mathematica, Tome 171 (2005) no. 3, pp. 239-260

Résumé

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.

Détail

DOI : 10.4064/sm171-3-3

Keywords: introduce study class locally convex vector lattices continuous functions locally compact hausdorff space which call regular vector lattices investigate general properties these spaces subspaces so called generalized affine functions moreover present korovkin type theorems continuous positive linear operators particular study korovkin subspaces finitely defined operators identity operator positive projections due its length paper split parts distinct character first part introduce class regular vector lattices prove integral representation theorem continuous positive linear forms study enveloping functions related continuous positive operator together corresponding space generalized affine functions finally obtain stone weierstrass type theorem second part which appear journal present korovkin type theorems together applications