A note on linear mappings between function spaces

Jan Baars

Commentationes Mathematicae Universitatis Carolinae (1993)

  • Volume: 34, Issue: 4, page 711-715
  • ISSN: 0010-2628

Abstract

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Arhangel’skiǐ 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, σ -compactness and realcompactness. In this paper we prove that if φ : C p ( X ) → C p ( X ) is a continuous linear surjection, then Y is pseudocompact provided X is and if φ is a continuous linear injection, then X is pseudocompact provided Y is. We also give examples that both statements do not hold for compactness, σ -compactness and realcompactness.

How to cite

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Baars, Jan. "A note on linear mappings between function spaces." Commentationes Mathematicae Universitatis Carolinae 34.4 (1993): 711-715. <http://eudml.org/doc/247486>.

@article{Baars1993,
abstract = {Arhangel’skiǐ 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.},
author = {Baars, Jan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {function space; topology of pointwise convergence},
language = {eng},
number = {4},
pages = {711-715},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A note on linear mappings between function spaces},
url = {http://eudml.org/doc/247486},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Baars, Jan
TI - A note on linear mappings between function spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 4
SP - 711
EP - 715
AB - Arhangel’skiǐ 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.
LA - eng
KW - function space; topology of pointwise convergence
UR - http://eudml.org/doc/247486
ER -

References

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  1. Arhangel'skiĭ A.V., On linear homeomorphisms of function spaces, Soviet Math. Dokl. 25 (1982), 852-855. (1982) 
  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. Zbl0755.54007MR1182148
  3. Baars J., de Groot J., Pelant J., Function spaces of completely metrizable spaces, to appear in Trans. of the AMS. Zbl0841.54012MR1182148

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