Distance to Spaces of Semicontinuous and Continuous Functions

Carlos Angosto

doi:10.1155/2019/9602504

Abstract

Given a topological spaceX, we establish formulas to compute the distance from a functionf∈RXto the spaces of upper semicontinuous functions and lower semicontinuous functions. For this, we introduce an index of upper semioscillation and lower semioscillation. We also establish new formulas about distances to some subspaces of continuous functions that generalize some classical results.

Highlights

Several classical and modern results deal with distances, optimization, equalities, and inequalities
Several authors have studied how to compute distances to some spaces of functions, for example, spaces of continuous functions ([1]), spaces of Baire-one functions ([2, 3]), and spaces of measurable functions and integrable functions ([4]). This kind of results has been used in a big number of papers
We prove the theorem in the usC(X) case, and the other one can be done analogously or we can deduce it from the usC(X) case applied to −f

Summary

Introduction

Several classical and modern results deal with distances, optimization, equalities, and inequalities. Several authors have studied how to compute distances to some spaces of functions, for example, spaces of continuous functions ([1]), spaces of Baire-one functions ([2, 3]), and spaces of measurable functions and integrable functions ([4]) This kind of results has been used in a big number of papers (for instance, [5,6,7,8,9,10,11,12,13,14,15]). Theorem 15 studies the distance from f to the continuous functions that have fixed values in some points

Upper and Lower Semioscillation

Distances to Subspaces of Continuous Functions