Extenders for vector-valued functions

Iryna Banakh, Taras Banakh, and Kaori Yamazaki. "Extenders for vector-valued functions." Studia Mathematica 191.2 (2009): 123-150. <http://eudml.org/doc/286529>.

@article{IrynaBanakh2009,
abstract = {Given a subset A of a topological space X, a locally convex space Y, and a family ℂ of subsets of Y we study the problem of the existence of a linear ℂ-extender $u: C_\{∞\}(A,Y) → C_\{∞\}(X,Y)$, which is a linear operator extending bounded continuous functions f: A → C ⊂ Y, C ∈ ℂ, to bounded continuous functions f̅ = u(f): X → C ⊂ Y. Two necessary conditions for the existence of such an extender are found in terms of a topological game, which is a modification of the classical strong Choquet game. The results obtained allow us to characterize reflexive Banach spaces as the only normed spaces Y such that for every closed subset A of a GO-space X there is a ℂ-extender $u: C_\{∞\}(A,Y) → C_\{∞\}(X,Y)$ for the family ℂ of closed convex subsets of Y. Also we obtain a characterization of Polish spaces and of weakly sequentially complete Banach lattices in terms of extenders.},
author = {Iryna Banakh, Taras Banakh, Kaori Yamazaki},
journal = {Studia Mathematica},
keywords = {linear extender; -extender; -extender; monotone extender; (countably) semireflexive locally convex space; reflexive Banach space; Polish space; weakly sequential complete Banach lattice; GO-space; strong Choquet game; Michael line},
language = {eng},
number = {2},
pages = {123-150},
title = {Extenders for vector-valued functions},
url = {http://eudml.org/doc/286529},
volume = {191},
year = {2009},
}

TY - JOUR
AU - Iryna Banakh
AU - Taras Banakh
AU - Kaori Yamazaki
TI - Extenders for vector-valued functions
JO - Studia Mathematica
PY - 2009
VL - 191
IS - 2
SP - 123
EP - 150
AB - Given a subset A of a topological space X, a locally convex space Y, and a family ℂ of subsets of Y we study the problem of the existence of a linear ℂ-extender $u: C_{∞}(A,Y) → C_{∞}(X,Y)$, which is a linear operator extending bounded continuous functions f: A → C ⊂ Y, C ∈ ℂ, to bounded continuous functions f̅ = u(f): X → C ⊂ Y. Two necessary conditions for the existence of such an extender are found in terms of a topological game, which is a modification of the classical strong Choquet game. The results obtained allow us to characterize reflexive Banach spaces as the only normed spaces Y such that for every closed subset A of a GO-space X there is a ℂ-extender $u: C_{∞}(A,Y) → C_{∞}(X,Y)$ for the family ℂ of closed convex subsets of Y. Also we obtain a characterization of Polish spaces and of weakly sequentially complete Banach lattices in terms of extenders.
LA - eng
KW - linear extender; -extender; -extender; monotone extender; (countably) semireflexive locally convex space; reflexive Banach space; Polish space; weakly sequential complete Banach lattice; GO-space; strong Choquet game; Michael line
UR - http://eudml.org/doc/286529
ER -