(DF)-Spaces of type
and ![C\overline{V}(X, E)](http://212.189.136.205/plugins/generic/latexRender/cache/be84fcc44d77e6dbd0c9404c344f9580.png)
Abstract
Some locally convex properties of the spaces
of the bounded continuous functions on a completely regular Hausdorff space X with values in a (DF-space) E are studied and applied to the (DF)-spaces of type
(e.g., see [S]).The following are our main results:
1.
is a (DF)-space if and only if E is a (DF)-space.
2.For a (DF)-space E,
is quasi barrelled if and only if either (i)X is pseudocompact and E is quasibarrelled or (ii) X is not pseudocompact and the bounded subsets of E are metrizaable.
3. If
and if each
is dominated by some
, then
(resp.,
) is a (DF)-space if and only if E is a (DF)-space.
4. Let X be a locally compact and σ-compact space,
and E a (DF)-space. Then
is quasibarrelled if and only if (i) E is quasibarrelled and
satisfies condition
or (ii) the bounded subsets of E are metrizable and
satisfies condition (D).
![CB( X, E)](http://212.189.136.205/plugins/generic/latexRender/cache/496aac1854d7fcd5090d8b8fe2b87022.png)
![C\bar{V}(X,E)](http://212.189.136.205/plugins/generic/latexRender/cache/06b8226ba2c064cac58edb38daad1d10.png)
![CB(X,E)](http://212.189.136.205/plugins/generic/latexRender/cache/165e806afe81cc91bf0cd29b4ce5299c.png)
![CB(X,E)](http://212.189.136.205/plugins/generic/latexRender/cache/165e806afe81cc91bf0cd29b4ce5299c.png)
![\mathcal V ⊂ C(X)](http://212.189.136.205/plugins/generic/latexRender/cache/7a0a544bd76b8c6aaf92bdeb92b33edd.png)
![\bar{v}∈\bar{V}](http://212.189.136.205/plugins/generic/latexRender/cache/efb25a33317a7759d23b2daaa3153c6d.png)
![\tilde{v}∈ \bar{V}∩ C(X)](http://212.189.136.205/plugins/generic/latexRender/cache/603eaa1c974ea49eba4ed30e364aa5cd.png)
![C\bar{V}(X,E)](http://212.189.136.205/plugins/generic/latexRender/cache/06b8226ba2c064cac58edb38daad1d10.png)
![C\bar{V}(X)⨂_\varepsilon E](http://212.189.136.205/plugins/generic/latexRender/cache/45cb121d70529e5e3697ed80ce467508.png)
![\mathcal V ⊂ C(X)](http://212.189.136.205/plugins/generic/latexRender/cache/7a0a544bd76b8c6aaf92bdeb92b33edd.png)
![C\bar{V}(X,E)](http://212.189.136.205/plugins/generic/latexRender/cache/06b8226ba2c064cac58edb38daad1d10.png)
![\mathcal V](http://212.189.136.205/plugins/generic/latexRender/cache/89135ae0eb5bcc1af22e8a6d910c39d8.png)
![( M, K)](http://212.189.136.205/plugins/generic/latexRender/cache/6b858177e6322d50206c6296ebd93aea.png)
![\mathcal V](http://212.189.136.205/plugins/generic/latexRender/cache/89135ae0eb5bcc1af22e8a6d910c39d8.png)
DOI Code:
10.1285/i15900932v10supn1p127
Full Text: PDF