Strong topologies on vector-valued function spaces

Marian Nowak

Czechoslovak Mathematical Journal (2000)

  • Volume: 50, Issue: 2, page 401-414
  • ISSN: 0011-4642

Abstract

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Let ( X , ∥ · ∥ X ) be a real Banach space and let E be an ideal of L 0 over a σ -finite measure space ( Ø , Σ , μ ) . Let ( X ) be the space of all strongly Σ -measurable functions f Ø → X such that the scalar function f ˜ , defined by f ˜ ( ø ) = ∥ f ( ø ) ∥ X for ø ∈ Ø , belongs to E . The paper deals with strong topologies on E ( X ) . In particular, the strong topology β ( E ( X ) , E ( X ) n ∼ ) ( E ( X ) n ∼ = the order continuous dual of E ( X ) ) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.

How to cite

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Nowak, Marian. "Strong topologies on vector-valued function spaces." Czechoslovak Mathematical Journal 50.2 (2000): 401-414. <http://eudml.org/doc/30571>.

@article{Nowak2000,
abstract = {Let $(X,\Vert \cdot \Vert _X)$ be a real Banach space and let $E$ be an ideal of $L^0$ over a $\sigma $-finite measure space $(Ø,\Sigma ,\mu )$. Let $(X)$ be the space of all strongly $\Sigma $-measurable functions $f\: Ø\rightarrow X$ such that the scalar function $\{\widetilde\{f\}\}$, defined by $\{\widetilde\{f\}\}(ø)=\Vert f(ø)\Vert _X$ for $ø\in Ø$, belongs to $E$. The paper deals with strong topologies on $E(X)$. In particular, the strong topology $\beta (E(X), E(X)^\sim _n)$ ($E(X)^\sim _n=$ the order continuous dual of $E(X)$) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.},
author = {Nowak, Marian},
journal = {Czechoslovak Mathematical Journal},
keywords = {vector valued function spaces; locally solid topologies; strong topologies; Mackey topologies; absolute weak topologies; vector valued function spaces; locally solid topologies; strong topologies; Mackey topologies; absolute weak topologies},
language = {eng},
number = {2},
pages = {401-414},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Strong topologies on vector-valued function spaces},
url = {http://eudml.org/doc/30571},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Nowak, Marian
TI - Strong topologies on vector-valued function spaces
JO - Czechoslovak Mathematical Journal
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 2
SP - 401
EP - 414
AB - Let $(X,\Vert \cdot \Vert _X)$ be a real Banach space and let $E$ be an ideal of $L^0$ over a $\sigma $-finite measure space $(Ø,\Sigma ,\mu )$. Let $(X)$ be the space of all strongly $\Sigma $-measurable functions $f\: Ø\rightarrow X$ such that the scalar function ${\widetilde{f}}$, defined by ${\widetilde{f}}(ø)=\Vert f(ø)\Vert _X$ for $ø\in Ø$, belongs to $E$. The paper deals with strong topologies on $E(X)$. In particular, the strong topology $\beta (E(X), E(X)^\sim _n)$ ($E(X)^\sim _n=$ the order continuous dual of $E(X)$) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.
LA - eng
KW - vector valued function spaces; locally solid topologies; strong topologies; Mackey topologies; absolute weak topologies; vector valued function spaces; locally solid topologies; strong topologies; Mackey topologies; absolute weak topologies
UR - http://eudml.org/doc/30571
ER -

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