The average distance property of classical Banach spaces II

The average distance property of classical Banach spaces II

A Banach space X has the average distance property (ADP) if there exists a unique real number r such that for each positive integer n and all x1, . . . , xn in the unit sphere of X there is some x in the unit sphere of X such that 1 n n ∑ k=1 ‖xk − x‖ = r. We show that lp does not have the average distance property if p > 2. This completes the study of the ADP for lp spaces.

On the Average Distance Property in Finite Dimensional Real Banach Spaces

The Banach-saks Property of the Banach Product Spaces

In this paper we first take a detail survey of the study of the Banach-Saks property of Banach spaces and then show the Banach-Saks property of the product spaces generated by a finite number of Banach spaces having the Banach-Saks property. A more general inequality for integrals of a class of composite functions is also given by using this property.

The Complete Continuity Property in Banach Spaces

Let X be a complex Banach space. We show that the following are equivalent: (i) X has the complete continuity property, (ii) for every (or equivalently for some) 1 < p < ∞, for f ∈ h(D, X) and rn ↑ 1, the sequence frn is p-Pettis-Cauchy, where frn is defined by frn(t) = f(rne ) for t ∈ [0, 2π], (iii) for every (or equivalently for some) 1 < p < ∞, for every μ ∈ V (X), the bounded linear operato...

Banach Spaces with the 2-summing Property

A Banach space X has the 2-summing property if the norm of every linear operator from X to a Hilbert space is equal to the 2-summing norm of the operator. Up to a point, the theory of spaces which have this property is independent of the scalar eld: the property is self-dual and any space with the property is a nite dimensional space of maximal distance to the Hilbert space of the same dimensio...