Some new properties of the stationary sets (defined by G. Pisier in [12]) are studied. Some arithmetical conditions are given, leading to the non-stationarity of the prime numbers. It is shown that any stationary set is a set of continuity. Some examples of "large" stationary sets are given, which are not sets of uniform convergence.

If every member of a class P of Banach spaces has a projectional resolution of the identity such that certain subspaces arising out of this resolution are also in the class P, then it is proved that every Banach space in P has a strong M-basis. Consequently, every weakly countably determined space, the dual of every Asplund space, every Banach space with an M-basis such that the dual unit ball is weak* angelic and every C(K) space for a Valdivia compact set K , has a strong M-basis.

It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space $X^{\xi }$ generated by subsequences $\left(e_{l_n}^{\xi }\right)$ and $\left(e_{m_n}^{\xi }\right)$, respectively, of the natural Schauder basis $\left(e_n^{\xi }\right)$ of $X^{\xi }$ are isomorphic if and only if $\left(e_{l_n}^{\xi }\right)$ and $\left(e_{m_n}^{\xi }\right)$ are equivalent. Further, $X^{\xi }$ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of $\left(e_n^{\xi }\right)$. It is also shown that there exists a complemented subspace spanned by a block basis of $\left(e_n^{\xi }\right)$, which is not isomorphic to a subspace generated by a subsequence of $\left(e_n^{\zeta }\right)$, for every $0\le \zeta \le \xi$....

We call a subset S of a topological vector space V linearly Borel if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V. It is shown that a Hamel basis of an infinite-dimensional Banach space can never be linearly Borel. This answers a question of Anatoliĭ Plichko.

Denote by F the composition operator generated by a given function f: R --&gt; R, acting on the space of absolutely continuous functions. In this paper we prove that the composition operator F maps the space AC[a,b] into itself if and only if f satisfies a local Lipschitz condition on R.

We study Kalton's theorem on the unconditional convergence of series of compact operators and we use some matrix techniques to obtain sufficient conditions, weaker than the previous one, on the convergence and unconditional convergence of series of compact operators.

In this note we investigate the relationship between the convergence of the sequence $\left\{S_n\right\}$ of sums of independent random elements of the form $S_n={\sum }_{i=1}^n{\epsilon }_i{x}_i$ (where ${\epsilon }_i$ takes the values $\pm 1$ with the same probability and $x_i$ belongs to a real Banach space $X$ for each $i\in \mathbb{N}$) and the existence of certain weakly unconditionally Cauchy subseries of ${\sum }_{n=1}^{\infty }{x}_n$.

We consider several greedy conditions for bases in Banach spaces that arise naturally in the study of the Thresholding Greedy Algorithm (TGA). In particular, we continue the study of almost greedy bases begun in [3]. We show that almost greedy bases are essentially optimal for n-term approximation when the TGA is modified to include a Chebyshev approximation. We prove that if a Banach space X has a basis and contains a complemented subspace with a symmetric basis and finite cotype then X has an...

A stronger version of the notion of frame in Banach space called Strong Retro Banach frame (SRBF) is defined and studied. It has been proved that if $𝒳$ is a Banach space such that ${𝒳}^{*}$ has a SRBF, then $𝒳$ has a Bi-Banach frame with some geometric property. Also, it has been proved that if a Banach space $𝒳$ has an approximative Schauder frame, then ${𝒳}^{*}$ has a SRBF. Finally, the existence of a non-linear SRBF in the conjugate of a separable Banach space has been proved.