Extenders for vector-valued functions

• Autorzy: Iryna Banakh , Taras Banakh , Kaori Yamazaki
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46A25: Reflexivity and semi-reflexivity
• 46A40: Ordered topological linear spaces, vector lattices
• 91A44: Games involving topology or set theory
• 91A80: Applications of game theory
• 46A55: Convex sets in topological linear spaces; Choquet theory
• 47B65: Positive operators and order-bounded operators
• 60B05: Probability measures on topological spaces
• 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
• 54C20: Extension of maps
• 46B42: Banach lattices
• 46B40: Ordered normed spaces
• 46B09: Probabilistic methods in Banach space theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2009
• Tom: 191
• Numer: 2
• Strony: 123-150

Daty

wydano

2009

Twórcy

autor

Iryna Banakh

• Department of Functional Analysis, Ya. Pidstryhach Institute for Applied Problems, of Mechanics and Mathematics, Naukova 3b, Lviv, Ukraine

autor

Taras Banakh

• Instytut Matematyki, Akademia Świętokrzyska, 25-406 Kielce, Poland
• Department of Mathematics, Ivan Franko National University of Lviv, Universytetska 1, 79000, Lviv, Ukraine

autor

Kaori Yamazaki

• Faculty of Economics, Takasaki City University of Economics, 1300 Kaminamie, Takasaki, Gunma 370-0801, Japan

Identyfikatory

DOI

10.4064/sm191-2-2