Let

Throughout the paper let

Assume that

Let

The strict topology

Throughout the paper we will assume that

For

Let

By

for each

Let

Assume that

Let

For

The following Bartle-Dunfor-Schwartz type theorem will be useful (see [

Let

Following [

A bounded linear operator

Note that

For each

For

The space of all

For the integration theory of functions

The following statements hold:

for

there exists a unique

and

for

In view of [

For a sequence

The following theorem gives a characterization of

Let

For each

For each

For each

For

For

Following [

We define

For each

The following result will be of importance (see [

Let

From Theorem

By

The following theorem characterizes

Let

For every

For each

As a consequence of Theorem

Making use of [

By

A completely regular Hausdorff space

From now on we will assume that

Assume that

(i)

(ii)

(iv)

Now we can state a characterization of

Let

For each

(i)

(ii)

(iii)

(iv)

In view of Theorem

Let

In view of Theorem

Assume that

Let

If

Let

Note that if

Recall that a function ^{*}-measurable functions

Following [

A bounded linear operator

Assume that

Since

Let ^{*}-measurable function

The function

For every

It follows that

Recall that a series

If

Let

For each

(i) Assume that

(ii) Let

Assume that

(i)

(ii)

Recall that a subset

Abbott et al. [

Assume that

(i)

(ii)

(iii)

Recall that a bounded linear operator

If

A bounded linear operator

Let

(i)

(ii)

From Proposition

Assume that

Let

For each

(i) It follows from Corollary

(ii) Let

Recall that a bounded linear operator

A bounded linear operator

Using Theorem

Let

Let

For each

(i) In view of [

(ii) Let

Assume that

(i)

(ii)

(iii)

(iv)

(v)

Assume that

(i)

If

Let

Assume that

Assume that

(i) In view of [

(ii) By Theorem

Assume that

As a consequence of Corollaries

Assume that

Following [

A

In particular, an operator

If

Let

Let

(i) Let

(ii) Let

For

Let

For each

(i) In view of [

(ii) Assume that

(iii) To show that

A bounded linear operator

Bilyeu and Lewis [

Now we show an analogue of Theorem 4.1 of [

Let

For each

Since

Let

The author declares that there is no conflict of interests regarding the publication of this paper.

_{o}(T, X)