Weak Integrals and Bounded Operators in Topological Vector Spaces

Abstract

Let X be a topological vector space and let S be a locally compact space. Let us consider the function space of all continuous functions , vanishing outside a compact set of S , equipped with an appropriate topology. In this work we will be concerned with the relationship between bounded operators , and X - valued integrals on . When X is a Banach space, such relation has been completely achieved via Bochner integral in [ 1 ]. In this paper we investigate the context of locally convex spaces and we will focus attention on weak integrals, namely the Pettis integrals. Some results in this direction have been obtained, under some special conditions on the structure of X and its topological dual X * . In this work we consider the case of a semi reflexive locally convex space and prove that each Pettis integral with respect to a signed measure μ , on S gives rise to a unique bounded operator , which has the given Pettis integral form.