Characterizing reflexive Banach spaces with help of extenders
by
Taras Banakh
Uniwersytet Humanistyczno-Przyrodniczy Jana Kochanowskiego w Kielcach, Ivan Franko National University of Lviv
Coauthors: Iryna Banakh, Kaori Yamazaki

We show that a normed space Y is reflexive if and only if for any closed subset A of a GO-space X there is a linear extension operator u:C_∞(A, Y)→ C_∞(X, Y) extending any bounded continuous functions f:A→ Y to a continuous function u(f):X→ Y such that the image u(f)(X) lies in the closed convex hull of f(A) in Y. This answers an old question of R.Heath and D.Lutzer [HL].

It is interesting to compare this result with the classical theorem of Dugundji saying that for each closed subset A of a metrizable space X and each locally convex space Y there is a linear extension operator u:C(A, Y)→ C(X, Y) extending any continuous function f:A→ Y to a continuous function u(f):X→ Y such that the image u(f)(X) lies in a convex hull of f(A) in Y.

More information of this topic can be found in [BBY].

References:

[BBY] I.Banakh, T.Banakh, K.Yamazaki, Extenders for vector-valued functions, Studia Math. (to appear).

[HL] R.W. Heath and D.J. Lutzer, Dugundji extension theorems for linearly ordered spaces, Pacific J. Math. 55 (1974), 419-425.

Date received: June 23, 2008