A characterization of Haar null sets in the sense of Christensen is given. Using it, we show that if the dual of a Banach space X has the Banach-Saks property, then closed and convex subsets of X with empty interior are Haar null.

We create a new family of Banach spaces, the James–Schreier spaces, by amalgamating two important classical Banach spaces: James’ quasi-reflexive Banach space on the one hand and Schreier’s Banach space giving a counterexample to the Banach–Saks property on the other. We then investigate the properties of these James–Schreier spaces, paying particular attention to how key properties of their ‘a...

ABanach space B is said to satisfy the Banach-Saks property with respect a regular summability method if every bounded subsequence has summable subsequence. We show that Banach satisfies Robison-Hamilton method, for double sequence there exists ?-subsequence whose subsequences are all same limit.

X iv :m at h/ 94 03 21 2v 1 [ m at h. FA ] 3 1 M ar 1 99 4 CLASSES OF BANACH SPACES STABLE AND UNSTABLE WITH RESPECT TO THE OPENING M.I.Ostrovskii Mathematics Department, Boǧaziçi University 80815 Bebek, Istanbul, TURKEY and Mathematical Division, Institute for Low Temperature Physics 47 Lenin avenue, 310164 Kharkov, UKRAINE e-mail: [email protected] Abstract. The paper is a complement...

In this note, we will discuss how to relate an operator ideal on Banach spaces to the sequential structures it defines. Concrete examples of ideals of compact, weakly compact, completely continuous, Banach-Saks and weakly Banach-Saks operators will be demonstrated.