A Banach space X is asymptotically symmetric (a.s.) if for some C < ∞, for all m ∈ ℕ, for all bounded sequences ${\left(x_j^i\right)}_{j=1}^{\infty }\subseteq X$, 1 ≤ i ≤ m, for all permutations σ of 1,...,m and all ultrafilters ₁,...,ₘ on ℕ, $li{m}_{n₁,₁}...li{m}_{nₘ,ₘ}\left|\right|{\sum }_{i=1}^m{x}_{n_i}^i\left|\right|\le Cli{m}_{n_{\sigma \left(1\right)}{,}_{\sigma \left(1\right)}}...li{m}_{n_{\sigma \left(m\right)}{,}_{\sigma \left(m\right)}}\left|\right|{\sum }_{i=1}^m{x}_{n_i}^i\left|\right|$. We investigate a.s. Banach spaces and several natural variations. X is weakly a.s. (w.a.s.) if the defining condition holds when restricted to weakly convergent sequences ${\left(x_j^i\right)}_{j=1}^{\infty }$. Moreover, X is w.n.a.s. if we restrict the condition further to normalized weakly null sequences. If X is a.s. then all spreading...

If ξ is a countable ordinal and (fk) a sequence of real-valued functions we define the repeated averages of order ξ of (fk). By using a partition theorem of Nash-Williams for families of finite subsets of positive integers it is proved that if ξ is a countable ordinal then every sequence (fk) of real-valued functions has a subsequence (f'k) such that either every sequence of repeated averages of order ξ of (f'k) converges uniformly to zero or no sequence of repeated averages of order ξ of (f'k)...

We give a characterization of compact spaces K such that the Banach space C(K) is isomorphic to the space c₀(Γ) for some set Γ. As an application we show that there exists an Eberlein compact space K of weight ${\omega }_{\omega }$ and with the third derived set $K^{\left(3\right)}$ empty such that the space C(K) is not isomorphic to any c₀(Γ). For this compactum K, the spaces C(K) and $c₀\left({\omega }_{\omega }\right)$ are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.

In this short note we give a negative answer to a question of Argyros, Castillo, Granero, Jiménez and Moreno concerning Banach spaces which contain complemented and uncomplemented subspaces isomorphic to c0.

A Banach space X is an M-ideal in its bidual if the relation ||f + w|| = ||f|| + ||w||holds for every f in X* and every w in X ⊥.The class of the Banach spaces which are M-ideals in their biduals, in short, the class of M-embedded spaces, has been carefully investigated, in particular by A. Lima, G. Godefroy and the West Berlin School. The spaces c0(I) -I any set- equipped with their canonical norm belong to this class, which also contains e.g. certain spaces K(E,F) of compact operators between...

The paper begins with a self-contained and short development of Bárány’s theorems of Carathéodory and Helly type in finite-dimensional spaces together with some new variants. In the second half the possible generalizations of these results to arbitrary Banach spaces are investigated. The Carathéodory-Bárány theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Bárány theorem reads as follows:...

We investigate various kinds of bases in infinite-dimensional Banach spaces. In particular, we consider the complexity of Hamel bases in separable and non-separable Banach spaces and show that in a separable Banach space a Hamel basis cannot be analytic, whereas there are non-separable Hilbert spaces which have a discrete and closed Hamel basis. Further we investigate the existence of certain complete minimal systems in ${\ell }_{\infty }$ as well as in separable Banach spaces.

Ogni spazio di Banach ha un sistema bibasico $(x_n,f_n)$ normalizzato; inoltre ogni successione $(x_n)$ uniformemente minimale appartiene ad un sistema biortogonale limitato $(x_n,f_n)$, dove $(f_n)$ è M-basica e normante.