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.

Many graph properties (e.g., connectedness, containing a complete subgraph) are known to be difficult to check. In a decision-tree model, the cost of an algorithm is measured by the number of edges in the graph that it queries. R. Karp conjectured in the early 1970s that all monotone graph properties are evasive—that is, any algorithm which computes a monotone graph property must check all edge...

whenever a belongs to the closed convex hull of the set of weak limit points of {an}∞n=1. In case X has the Banach-Saks property and {an}∞n=1 is bounded the converse assertion holds too. A characterization of reflexive spaces in terms of limit points and cores of bounded sequences is also given. The motivation for the problems investigated goes back to Lévy laplacian from potential theory in Hi...

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...

Let E be a rearrangement-invariant Banach function space on [0, 1], and let Γ(E) denote the set of all p ≥ 1 such that any sequence {xn} in E converging weakly to 0 has a subsequence {yn} with supm m−1/p‖ ∑ 1≤k≤m yn‖ < ∞. The set Γi(E) is defined similarly, but only sequences {xn} of independent random variables are taken into account. It is proved (under the assumption Γ(E) = {1}) that if Γi(E...

Introduced by Vitali(1907), Hahn(1922) and Saks(1933), the VitaliHahn-Saks theorem plays a vital role in measure theory in proving some weak convergence theorems. In this Toolbox Essay, we will first introduce the Nikodym theorem followed by a self-contained proof. Next, we state the Vitali-Hahn-Saks theorem and show by an example how this theorem can be used in probability theory. The benefit ...