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.

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

ALTOMARE, Francesco;CAPPELLETTI MONTANO, MIRELLA
2005-01-01

Abstract

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.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11586/17641
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 9
  • ???jsp.display-item.citation.isi??? 9
social impact