Weak Integrals and Bounded Operators in Topological Vector Spaces ()
1. 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
vanishing outside a compact set of S, put
if X = R. We are interested in representing linear bounded operators
, by means of weak integrals against scalar measures on the Borel
-field BS of S. Before handling more closely this problem, we need some topological facts about the space
.
If K is a compact set in S, let
be the set of all continuous functions
, vanishing outside K. It is clear that
is a linear subspace of
. We equip
with the topology
generated by the family of seminorms:
![](https://www.scirp.org/html/6-5300484\fa5c770a-d3b6-4400-b83e-364f8bc2a5d2.jpg)
where
is the family of seminorms generating the locally convex topology of X. The topology
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 K1 is a subset of K2, then the natural embedding
is continuous. This allows one to provide the space
with the inductive topology
induced by the subspaces
,
. The facts we need about the space,
is well known:
1.1. Proposition
1) The space
,
is locally convex Hausdorff and for each compact K, the relative topology of
on
is
, this means that the canonical embedding
is continuous.
2) Let
be a linear operator of
into the locally convex Hausdorff space V, then T is continuous if and only if the restriction
of T to the subspace
is continuous for each compact K.
1.2. Definition
For each
in the topological dual
of X and for each function
, define the function
on S by
. Then
sends
into
. Recall that
is equipped with the uniform norm.
1.3. Lemma
The operator
is linear and bounded. Moreover for each
,
is onto.
Proof: First it is clear that
. Now by Proposition 1.1(b), we have to show that for each compact set K of S the operator
is bounded. Since
is bounded, there is a seminorm
on X and a constant M such that
for all
. So we have
if
, and
,
; it follows that
.
Since by Formula (*), the right side of this inequality is
, we deduce that
is continuous. Now suppose
. Then there exists
such that
and
. It is clear that we can assume
. Now let
and define
by
, then
and we have
, because
. It follows that
is onto. ■
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].
1.4. Definition
We say that a bounded operator
has a Pettis integral form if there exists a scalar measure of bounded variation
on BS such that, for every continuous functional
in
, we have:
![](https://www.scirp.org/html/6-5300484\5cbc8178-f0fb-4c27-9d20-19545369f611.jpg)
See Reference [3] for details on Pettis integral.
2. 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].
2.1. Definition
Let P be the class of all bounded operators
satisfying the following condition:
(I) For
and
, if
then
.
It is easy to check that P is a subspace of the space
of all bounded operators from
to X. Also one can prove that P is closed in the weak operator topology of
. 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 BS. This is the content of the following theorem proved in [4].
2.2. Theorem
Let
be in the class P. Then there is a unique bounded signed measure
on BS such that
holds for all
in
and
. Moreover for each seminorm
on X we have
, where
is the total variation of
and
is the
-norm of T defined by
![](https://www.scirp.org/html/6-5300484\d7321d99-7b80-4479-8407-72e25b0d1931.jpg)
with
.
By this theorem we may denote each operator T in the class P by the conventional symbol
![](https://www.scirp.org/html/6-5300484\e46c4177-aabc-4dd5-b333-0b49b990388c.jpg)
where the letter P stands for Pettis integral.
3. 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.
3.1. Operators via Pettis Integrals
A little inspection of (W) suggests the following quite plausible observations: First the integral
, as a linear functional of
on
, should beat least continuous for some convenient topology on
Also the existence of the corresponding Tf in (W) will require that such topology on X should be compatible for the dual pair
. Finally, to get the continuity of the functional
, one can seek conditions such that if
in an appropriate manner, then
goes to 0 uniformly for
. Since
is bounded this will give
.
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).
3.2. Theorem
Let X be a locally convex space with the convex compactness property, and whose dual
is equipped with the Mackey topology
If
is a bounded scalar measure on BS, then there is a unique bounded operator
in the class P satisfying (W), with
for each seminorm
on X.
3.3. Theorem
Let X be a locally convex Hausdorff space whose dual
is a barrelled space. If
is a bounded signed measure on BS, 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.
3.4. Definition
The strong topology
of
is the topology generated by the family of the seminorms:
![](https://www.scirp.org/html/6-5300484\804a86e8-1797-41d1-bd95-cab4c06ee4f2.jpg)
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
, which is the topology of simple convergence on X. We shall denote by
the space
equipped with the
-topology (the
- topology). Then we have:
3.5. Proposition
1) For each
there exists a unique ![](https://www.scirp.org/html/6-5300484\a8334089-1103-4be5-9308-63ec7d10ada5.jpg)
such that:
.
2)
, that is, every weak * continuous functional on
is strongly continuous .
3.6. Definition
We say that the space X is semireflexive if
.
Now we are in a position to state the main results of this paper.
3.7. 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:
![](https://www.scirp.org/html/6-5300484\6cf66e82-f853-41fe-8708-87006f861012.jpg)
.
where
is the variation of
.
Proof: Fix f in
and define the functional
, by
. It is clear that
is linear. Moreover
. Indeed it is enough to prove that
. If
, in
, then for each bounded subset B of X,
uniformly for
. But since
, the set
is bounded, so
uniformly in
. Therefore,
, because the measure μ is of bounded variation. Hence
.
Since X is semireflexive,
; by Proposition 3.5(a), there is a unique
such that
,
. Now let us define the operator
by
,
. 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
be a seminorm on X, and let K be a compact subset of X. For
, we have:
![](https://www.scirp.org/html/6-5300484\28c043d9-3655-4163-a148-ae08e7912d63.jpg)
which proves the continuity of T.
Now to compute
, observe from the integral form of
that
.
Taking the supremum in both sides over
, the polar set of the unit ball
of X, we get:
![](https://www.scirp.org/html/6-5300484\de8d5329-8cbc-4e54-a717-9841b2240fed.jpg)
So we deduce that
. To see the reverse inequality, let us consider a function
of the form
, with
satisfying
and x fixed in X such that
. With this choice, the function f belongs to the unit ball
. Then we have
![](https://www.scirp.org/html/6-5300484\d9637c2d-0c51-4ab9-af7e-7f364293eccd.jpg)
and
![](https://www.scirp.org/html/6-5300484\b6f8960b-61c9-4aea-a3b2-ef53a8a03127.jpg)
so that
![](https://www.scirp.org/html/6-5300484\5685d8ad-a197-49f5-99e1-3aaf99871175.jpg)
since
. So we get
![](https://www.scirp.org/html/6-5300484\99871107-ca54-46bb-baf8-eeef56c130de.jpg)
because
.
Therefore
■
By appealing to theorem 2.3, we get the following rather precise theorem:
3.8. Theorem
Let 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
on BS. This correspondence is given by the relation
:
![](https://www.scirp.org/html/6-5300484\eaa6c46d-ff37-4a67-b506-d2dba4d8a015.jpg)
4. Acknowledgements
This work has been done under the Project No. 121/130/ 1432. The authors are grateful to the Deanship of Scientific Research of the King Abdulaziz University, Jeddah, for their financial support.