Locally solid topologies on spaces of vector-valued continuous functions

Nowak, Marian, and Rzepka, Aleksandra. "Locally solid topologies on spaces of vector-valued continuous functions." Commentationes Mathematicae Universitatis Carolinae 43.3 (2002): 473-483. <http://eudml.org/doc/248981>.

@article{Nowak2002,
abstract = {Let $X$ be a completely regular Hausdorff space and $E$ a real normed space. We examine the general properties of locally solid topologies on the space $C_b(X,E)$ of all $E$-valued continuous and bounded functions from $X$ into $E$. The mutual relationship between locally solid topologies on $C_b(X,E)$ and $C_b(X)$$(=C_b(X,\mathbb \{R\}))$ is considered. In particular, the mutual relationship between strict topologies on $C_b(X)$ and $C_b(X,E)$ is established. It is shown that the strict topology $\beta _\sigma (X,E)$ (respectively $\beta _\tau (X,E)$) is the finest $\sigma $-Dini topology (respectively Dini topology) on $C_b(X,E)$. A characterization of $\sigma $-Dini and Dini topologies on $C_b(X,E)$ in terms of their topological duals is given.},
author = {Nowak, Marian, Rzepka, Aleksandra},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {vector-valued continuous functions; strict topologies; locally solid topologies; Dini topologies; vector-valued continuous functions; strict topologies; locally solid topologies; Dini topologies},
language = {eng},
number = {3},
pages = {473-483},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Locally solid topologies on spaces of vector-valued continuous functions},
url = {http://eudml.org/doc/248981},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Nowak, Marian
AU - Rzepka, Aleksandra
TI - Locally solid topologies on spaces of vector-valued continuous functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 3
SP - 473
EP - 483
AB - Let $X$ be a completely regular Hausdorff space and $E$ a real normed space. We examine the general properties of locally solid topologies on the space $C_b(X,E)$ of all $E$-valued continuous and bounded functions from $X$ into $E$. The mutual relationship between locally solid topologies on $C_b(X,E)$ and $C_b(X)$$(=C_b(X,\mathbb {R}))$ is considered. In particular, the mutual relationship between strict topologies on $C_b(X)$ and $C_b(X,E)$ is established. It is shown that the strict topology $\beta _\sigma (X,E)$ (respectively $\beta _\tau (X,E)$) is the finest $\sigma $-Dini topology (respectively Dini topology) on $C_b(X,E)$. A characterization of $\sigma $-Dini and Dini topologies on $C_b(X,E)$ in terms of their topological duals is given.
LA - eng
KW - vector-valued continuous functions; strict topologies; locally solid topologies; Dini topologies; vector-valued continuous functions; strict topologies; locally solid topologies; Dini topologies
UR - http://eudml.org/doc/248981
ER -