Duality and some topological properties of vector-valued function spaces

Krzysztof Feledziak. "Duality and some topological properties of vector-valued function spaces." Commentationes Mathematicae 48.1 (2008): null. <http://eudml.org/doc/291464>.

@article{KrzysztofFeledziak2008,
abstract = {Let $E$ be an ideal of $L^0$ over $\sigma $-finite measure space $(\Omega , \Sigma , \mu )$ and let $(X, \Vert \cdot \Vert _X)$ be a real Banach space. Let $E(X)$ be a subspace of the space $L^0(X)$ of $\mu $-equivalence classes of all strongly $\Sigma $-measurable functions $f\colon \Omega \rightarrow X$ and consisting of all those $f\in L^0(X)$, for which the scalar function $\tilde\{f\} = \Vert f (\cdot )\Vert _X$ belongs to $E$. Let $E$ be equipped with a Hausdorff locally convex-solid topology $\xi $ and let $\xi $ stand for the topology on $E(X)$ associated with $\xi $. We examine the relationship between the properties of the space $(E(X), \xi )$ and the properties of both the spaces $(E, \xi )$ and $(X, \Vert · \Vert _X)$. In particular, it is proved that $E(X)$ (embedded in a natural way) is an order closed ideal of its bidual iff $E$ is an order closed ideal of its bidual and $X$ is reflexive. As an application, we obtain that $E(X)$ is perfect iff $E$ is perfect and $X$ is reflexive.},
author = {Krzysztof Feledziak},
journal = {Commentationes Mathematicae},
keywords = {vector-valued function spaces; locally solid topologies; KB-spaces; Levy topologies; Lebesgue topologies; order dual; order continuous dual; perfectness},
language = {eng},
number = {1},
pages = {null},
title = {Duality and some topological properties of vector-valued function spaces},
url = {http://eudml.org/doc/291464},
volume = {48},
year = {2008},
}

TY - JOUR
AU - Krzysztof Feledziak
TI - Duality and some topological properties of vector-valued function spaces
JO - Commentationes Mathematicae
PY - 2008
VL - 48
IS - 1
SP - null
AB - Let $E$ be an ideal of $L^0$ over $\sigma $-finite measure space $(\Omega , \Sigma , \mu )$ and let $(X, \Vert \cdot \Vert _X)$ be a real Banach space. Let $E(X)$ be a subspace of the space $L^0(X)$ of $\mu $-equivalence classes of all strongly $\Sigma $-measurable functions $f\colon \Omega \rightarrow X$ and consisting of all those $f\in L^0(X)$, for which the scalar function $\tilde{f} = \Vert f (\cdot )\Vert _X$ belongs to $E$. Let $E$ be equipped with a Hausdorff locally convex-solid topology $\xi $ and let $\xi $ stand for the topology on $E(X)$ associated with $\xi $. We examine the relationship between the properties of the space $(E(X), \xi )$ and the properties of both the spaces $(E, \xi )$ and $(X, \Vert · \Vert _X)$. In particular, it is proved that $E(X)$ (embedded in a natural way) is an order closed ideal of its bidual iff $E$ is an order closed ideal of its bidual and $X$ is reflexive. As an application, we obtain that $E(X)$ is perfect iff $E$ is perfect and $X$ is reflexive.
LA - eng
KW - vector-valued function spaces; locally solid topologies; KB-spaces; Levy topologies; Lebesgue topologies; order dual; order continuous dual; perfectness
UR - http://eudml.org/doc/291464
ER -