Extenders for vector-valued functions

Iryna Banakh ¹   ; Taras Banakh ²   ; Kaori Yamazaki ³

¹ Department of Functional Analysis Ya. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics Naukova 3b, Lviv, Ukraine
² Instytut Matematyki Akademia Świętokrzyska 25-406 Kielce, Poland and Department of Mathematics Ivan Franko National University of Lviv Universytetska 1 79000, Lviv, Ukraine
³ Faculty of Economics Takasaki City University of Economics 1300 Kaminamie, Takasaki Gunma 370-0801, Japan

Studia Mathematica, Tome 191 (2009) no. 2, pp. 123-150

Résumé

Given a subset $A$ of a topological space $X$, a locally convex space $Y$, and a family $\mathbb C$ of subsets of $Y$ we study the problem of the existence of a linear $\mathbb C$-extender $u:C_\infty(A,Y)\to C_\infty(X,Y)$, which is a linear operator extending bounded continuous functions $f:A\to C\subset Y$, $C\in\mathbb C$, to bounded continuous functions $\overline f =u(f):X\to C\subset Y$. 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 $\mathbb C$-extender $u:C_\infty(A,Y)\to C_\infty(X,Y)$ for the family $\mathbb C$ 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.

Détail

DOI : 10.4064/sm191-2-2

Keywords: given subset topological space locally convex space nbsp family mathbb subsets study problem existence linear mathbb c extender infty infty which linear operator extending bounded continuous functions subset mathbb bounded continuous functions overline subset necessary conditions existence extender found terms topological game which modification classical strong choquet game results obtained allow characterize reflexive banach spaces only normed spaces every closed subset go space there mathbb c extender infty infty family mathbb closed convex subsets obtain characterization polish spaces weakly sequentially complete banach lattices terms extenders