| Title:
|
A note on linear mappings between function spaces (English) |
| Author:
|
Baars, Jan |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
34 |
| Issue:
|
4 |
| Year:
|
1993 |
| Pages:
|
711-715 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Arhangel'ski\v{\i} proved that if $X$ and $Y$ are completely regular spaces such that ${C_p (X)}$ and ${C_p (Y)}$ are linearly homeomorphic, then $X$ is pseudocompact if and only if $Y$ is pseudocompact. In addition he proved the same result for compactness, $\sigma $-compactness and realcompactness. In this paper we prove that if $\phi : {C_p (X)} \rightarrow {C_p (X)}$ is a continuous linear surjection, then $Y$ is pseudocompact provided $X$ is and if $\phi $ is a continuous linear injection, then $X$ is pseudocompact provided $Y$ is. We also give examples that both statements do not hold for compactness, $\sigma $-compactness and realcompactness. (English) |
| Keyword:
|
function space |
| Keyword:
|
topology of pointwise convergence |
| MSC:
|
54C35 |
| MSC:
|
57N17 |
| idZBL:
|
Zbl 0787.54017 |
| idMR:
|
MR1263800 |
| . |
| Date available:
|
2009-01-08T18:07:38Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118628 |
| . |
| Reference:
|
[1] Arhangel'skiĭ A.V.: On linear homeomorphisms of function spaces.Soviet Math. Dokl. 25 (1982), 852-855. |
| Reference:
|
[2] Baars J., de Groot J.: On Topological and Linear Equivalence of Certain Function Spaces.CWI-tract 86, Centre for Mathematics and Computer Science, Amsterdam. Zbl 0755.54007, MR 1182148 |
| Reference:
|
[3] Baars J., de Groot J., Pelant J.: Function spaces of completely metrizable spaces.to appear in Trans. of the AMS. Zbl 0841.54012, MR 1182148 |
| . |
