Restricted uniform boundedness in Banach spaces

Abstract

Precise conditions for a subset A of a Banach space X are known in order that pointwise bounded on A sequences of bounded linear functionals on X are uniformly bounded. In this paper, we study such conditions under the extra assumption that the functionals belong to a given linear subspace Γ of X*. When Γ = X*, these conditions are known to be the same ones assuring a bounded linear operator into X, having A in its image, to be onto. We prove that, for A, deciding uniform boundedness of sequences in Γ is the same property as deciding surjectivity for certain classes of operators.

Keywords: Uniform boundedness; thick set; boundedness deciding set

Quaestiones Mathematicae 32(2009), 5–14