Duality theory of spaces of vector-valued continuous functions

Nowak, Marian, and Rzepka, Aleksandra. "Duality theory of spaces of vector-valued continuous functions." Commentationes Mathematicae Universitatis Carolinae 46.1 (2005): 55-73. <http://eudml.org/doc/249538>.

@article{Nowak2005,
abstract = {Let $X$ be a completely regular Hausdorff space, $E$ a real normed space, and let $C_b(X,E)$ be the space of all bounded continuous $E$-valued functions on $X$. We develop the general duality theory of the space $C_b(X,E)$ endowed with locally solid topologies; in particular with the strict topologies $\beta _z(X,E)$ for $z=\sigma , \tau , t$. As an application, we consider criteria for relative weak-star compactness in the spaces of vector measures $M_z(X,E^\{\prime \})$ for $z=\sigma , \tau , t$. It is shown that if a subset $H$ of $M_z(X,E^\{\prime \})$ is relatively $\sigma (M_z(X,E^\{\prime \}), C_b(X,E))$-compact, then the set $\operatorname\{conv\} (S(H))$ is still relatively $\sigma (M_z(X,E^\{\prime \}), C_b(X,E))$-compact ($S(H)=$ the solid hull of $H$ in $M_z(X,E^\{\prime \})$). A Mackey-Arens type theorem for locally convex-solid topologies on $C_b(X,E)$ is obtained.},
author = {Nowak, Marian, Rzepka, Aleksandra},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {vector-valued continuous functions; strict topologies; locally solid topologies; weak-star compactness; vector measures; strict topologies; locally solid topologies; weak-star compactness; vector measures},
language = {eng},
number = {1},
pages = {55-73},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Duality theory of spaces of vector-valued continuous functions},
url = {http://eudml.org/doc/249538},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Nowak, Marian
AU - Rzepka, Aleksandra
TI - Duality theory of spaces of vector-valued continuous functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 1
SP - 55
EP - 73
AB - Let $X$ be a completely regular Hausdorff space, $E$ a real normed space, and let $C_b(X,E)$ be the space of all bounded continuous $E$-valued functions on $X$. We develop the general duality theory of the space $C_b(X,E)$ endowed with locally solid topologies; in particular with the strict topologies $\beta _z(X,E)$ for $z=\sigma , \tau , t$. As an application, we consider criteria for relative weak-star compactness in the spaces of vector measures $M_z(X,E^{\prime })$ for $z=\sigma , \tau , t$. It is shown that if a subset $H$ of $M_z(X,E^{\prime })$ is relatively $\sigma (M_z(X,E^{\prime }), C_b(X,E))$-compact, then the set $\operatorname{conv} (S(H))$ is still relatively $\sigma (M_z(X,E^{\prime }), C_b(X,E))$-compact ($S(H)=$ the solid hull of $H$ in $M_z(X,E^{\prime })$). A Mackey-Arens type theorem for locally convex-solid topologies on $C_b(X,E)$ is obtained.
LA - eng
KW - vector-valued continuous functions; strict topologies; locally solid topologies; weak-star compactness; vector measures; strict topologies; locally solid topologies; weak-star compactness; vector measures
UR - http://eudml.org/doc/249538
ER -