Abstract
Let
X be a locally convex Hausdorff space and let
C
0
(
S
,
X
)
be the space of all continuous functions
f
:
S
→
X
, with compact support on the locally compact space
S. In this paper we prove a Riesz representation theorem for a class of bounded operators
T
:
C
0
(
S
,
X
)
→
X
, where the representing integrals are
X-valued Pettis integrals with respect to bounded signed measures on
S. Under the additional assumption that
X is a locally convex space, having the convex compactness property, or either,
X is a locally convex space whose dual
X
′
is a barrelled space for an appropriate topology, we obtain a complete identification between all
X-valued Pettis integrals on
S and the bounded operators
T
:
C
0
(
S
,
X
)
→
X
they represent. Finally we give two illustrations of the representation theorem proved, in the particular case when
X is the topological dual of a locally convex space.