Weak Integrals and Bounded Operators in Topological Vector Spaces

Let X be a topological vector space and let S be a locally compact space. Let us consider the function space   0 , C S X of all continuous functions : f S X  , 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   0 : , T C S X X  , 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  0 , C S X  * 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  , on S gives rise to a unique bounded operator , which has the given Pettis integral form.   X X  0 : , T C S

of all continuous functions : f S X  , 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   0 : , T C S X X  , 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  0 , C S X  * 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  , on S gives rise to a unique bounded operator , which has the given Pettis integral form.

Topological Preliminaries
Suppose that S is a locally compact space and let X be a locally convex TVS.We denote by the set of all continuous functions  0 , vanishing outside a compact set of S, put 0 0 if X = R.We are interested in representing linear bounded operators , by means of weak integrals against scalar measures on the Borel Before handling more closely this problem, we need some topological facts about the space .

 
with the topology K  generated by the family of seminorms: where   p  is the family of seminorms generating the locally convex topology of X.The topology K  is the topology of uniform convergence on K.
Next let us observe that , the union being performed over all the compact subsets K of S. On the other hand if K 1 is a subset of K 2 , then the natural embedding 1 2 is continuous.This allows one to provide the space : , , , ,

Lemma
The operator U  is linear and bounded.Moreover for each 0 . Now by Proposition 1.1(b), we have to show that for each compact set K of S the operator Since by Formula (*), the right side of this inequality is and we have ■ Now we consider the relationship between bounded operators , and weak integrals in the sense of the following definition.Such relationship is reminiscent to the classical Riesz theorem [2].

Definition
We say that a bounded operator has a Pettis integral form if there exists a scalar measure of bounded variation  on B S such that, for every continuous functional  in * X , we have: See Reference [3] for details on Pettis integral.

Integral Representation by Pettis Integral
In what follows, we introduce a class of bounded operators , which is, in this context, similar to the class used in [1].

Definition
Let P be the class of all bounded operators satisfying the following condition: . Note also that for a given bounded , Definition 1.4 implies condition (I) i.e.  .The crucial point is that condition (I) implies the Pettis integral form of Definition 1.4, for some bounded scalar measure  on B S .This is the content of the following theorem proved in [4].
By this theorem we may denote each operator T in the class P by the conventional symbol where the letter P stands for Pettis integral.

Operators Associated to Scalar Measures via Pettis Integrals
In this section we start with a bounded scalar measure  on and we seek for a linear bounded such that the correspondence between  and T would be given by formula (W).First let us make some observations.

Operators via Pettis Integrals
A little inspection of (W) suggests the following quite plausible observations: First the integral X , should beat least continuous for some convenient topology on * X Also the existence of the corresponding Tf in (W) will require that such topology on X should be compatible for the dual pair * , X X .Finally, to get the continuity of the functional 0. Such a program has been realized in [4], for a locally convex space having the convex compactness property [5], according to the following theorems (see [4] for details).

Theorem
Let X be a locally convex space with the convex compactness property, and whose dual * X is equipped with the Mackey topology   * , .X X


If  is a bounded scalar measure on B S , then there is a unique bounded operator in the class P satisfying (W), with for each seminorm p  on X.

Theorem
Let X be a locally convex Hausdorff space whose dual * X  is a barrelled space.If  is a bounded signed measure on B S , then there is a unique bounded operator in the class P satisfying (W) with respect to  and such that Most of these results have been obtained for a space whose dual is a Mackey space.It is natural to ask if similar representations can be established if the dual is endowed with another topology, e.g. the strong topology.

Definition
The strong topology   * , X X  of * X is the topology generated by the family of the seminorms: where B is running over all the bounded sets of X.
It is the topology of uniform convergence on the bounded sets of X.When we restrict to the finite sets B of X we get the so called weak * topology functional on X  is strongly continuous .

Definition
We say that the space X is semireflexive if Now we are in a position to state the main results of this paper.

Theorem
Let X be a locally convex Hausdorff semireflexive space.
If  is a bounded signed measure on , then there is a unique bounded operator in the class P satisfying: where  is the variation of  .
Proof: Fix f in   0 , C S X and define the functional Since X is semireflexive, ; by Proposition 3.5(a), there is a unique . It is easily checked that T is linear, and satisfies the condition of the theorem by construction.We have to show that T is bounded.Let p  be a seminorm on X, and let K be a compact subset of X.For

 
, , f C S K X  , we have: Sup Sup , . .
Taking the supremum in both sides over Sup Sup , ( ) .

Sup for
with the inductive topology  induced by the subspaces , , C S K X , K  .The facts we need about the space, ,  is locally convex Hausdorff and for each compact K, the relative topology of  on X  X be in the class P. Then there is a unique bounded signed measure  on B S such that  is the total variation of  and p T  is the p  -norm of T defined by which is the topology of simple convergence on X.We shall denote by 

■
see the reverse inequality, let us consider a function this choice, the function f belongs to the unit ball p B  By appealing to theorem 2.3, we get the following rather precise theorem:3.8.TheoremLet X be a locally convex Hausdorff semireflexive space.Then there is a one to one correspondence between the bounded operators of the class P and the X-valued Pettis integrals with respect to some bounded signed measure