Extenders for vector-valued functions

Tom 191 / 2009

Iryna Banakh, Taras Banakh, Kaori Yamazaki Studia Mathematica 191 (2009), 123-150 MSC: 46A25, 46A40, 46A55, 46B09, 46B40, 46B42, 47B65, 54C20, 54F05, 60B05, 91A44, 91A80. DOI: 10.4064/sm191-2-2


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.


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

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