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.

We investigate the bounded Ciesielski systems, which can be obtained from the spline systems of order (m,k) in the same way as the Walsh system arises from the Haar system. It is shown that the maximal operator of the Fejér means of the Ciesielski-Fourier series is bounded from the Hardy space $H_p$ to $L_p$ if 1/2 < p < ∞ and m ≥ 0, |k| ≤ m + 1. Moreover, it is of weak type (1,1). As a consequence, the Fejér means of the Ciesielski-Fourier series of a function f converges to f a.e. if f ∈ L₁ as n...

In one of the earliest monographs that involve the notion of a Schauder basis, Franklin showed that the Gram-Schmidt orthonormalization of a certain Schauder basis for the Banach space of functions continuous on [0,1] is again a Schauder basis for that space. Subsequently, Ciesielski observed that the Gram-Schmidt orthonormalization of any Schauder system is a Schauder basis not only for C[0,1], but also for each of the spaces $L^p[0,1]$, 1 ≤ p < ∞. Although perhaps not probable, the latter result would...

We consider the question of whether the trigonometric system can be equivalent to some rearrangement of the Walsh system in $L_p$ for some p ≠ 2. We show that this question is closely related to a combinatorial problem. This enables us to prove non-equivalence for a number of rearrangements. Previously this was known for the Walsh-Paley order only.

It is shown that every uncountable symmetric basic set in an F-space with a symmetric basis is equivalent to a basic set generated by one vector. We apply this result to investigate the structure of uncountable symmetric basic sets in Orlicz and Lorentz spaces.

For a Banach space X with an unconditional basic sequence, one of the following regular-irregular alternatives holds: either X contains a subspace isomorphic to ℓ₂, or X contains a subspace which has an unconditional finite-dimensional decomposition, but does not admit such a decomposition with a uniform bound for the dimensions of the decomposition. This result can be viewed in the context of Gowers' dichotomy theorem.

We prove that there exists an absolute constant c such that for any positive integer n and any system Φ of $2^n$ characters of a compact abelian group, $2^{-n/2}{t}_{\Phi }\left(T\right)\le c{n}^{-1/2}{t}_n\left(T\right)$, where T is an arbitrary operator between Banach spaces, $t_{\Phi }\left(T\right)$ is the type norm of T with respect to Φ and $t_n\left(T\right)$ is the usual Rademacher type-2 norm computed with n vectors. For the system of the first $2^n$ Walsh functions this is even true with c=1. This result combined with known properties of such type norms provides easy access to quantitative versions of...

In this note we show that if the ratio of the minimal volume V of n-dimensional parallelepipeds containing the unit ball of an n-dimensional real normed space X to the maximal volume v of n-dimensional crosspolytopes inscribed in this ball is equal to n!, then the relation of orthogonality in X is symmetric. Hence we deduce the following properties: (i) if V/v=n! and if n>2, then X is an inner product space; (ii) in every finite-dimensional normed space there exist at least two different Auerbach...