TITLE:
Weak Integrals and Bounded Operators in Topological Vector Spaces
AUTHORS:
Lakhdar Meziani, Saud M. Alsulami
KEYWORDS:
Bounded Operators; Integral Representation; Pettis Integral
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.3 No.5,
August
2,
2013
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.