Related papers: On $(p,r)$-null sequences and their relatives

We consider a normalized basis in a Banach space with the following property: any normalized block sequence of the basis has a subsequence equivalent to the basis. We show that under uniformity or other natural assumptions, a basis with…

Let $f$ be a continuous monotone real function defined on a compact interval $[a,b]$ of the real line. Given a sequence of partitions of $[a,b]$, $% \Delta_n $, $\left\Vert {\Delta }_{n}\right\Vert \rightarrow 0$, and given $l\geq 0,m\geq…

Let $X$ be a Banach space and $\mu$ a probability measure. A set $K \subseteq L^1(\mu,X)$ is said to be a $\delta\mathcal{S}$-set if it is uniformly integrable and for every $\delta>0$ there is a weakly compact set $W \subseteq X$ such that…

Let $X_0, X_1, ..., X_k$ with $k \in \IN\cup\{\infty\}$ be sequence spaces $($finite or infinite dimensional$)$ over $\IC$ or $\IR$ with absolute norms $N_i$ for $i = 0, ..., k$, $($i.e., with 1-unconditional bases$)$ such that $\dim X_0 =…

Given a Banach space $X$, for $n\in \mathbb N$ and $p\in (1,\infty)$ we investigate the smallest constant $\mathfrak P\in (0,\infty)$ for which every $f_1,...,f_n:{-1,1}^n\to X$ satisfy \int_{{-1,1}^n}\Bigg|\sum_{j=1}^n…

An ordered pair of semi-infinite binary sequences $(\eta,\xi)$ is said to be compatible if there is a way of removing a certain number (possibly infinite) of ones from $\eta$ and zeroes from $\xi$, whichwould map both sequences to the same…

Let X be a Banach space. Suppose that for all $p\in (1, \infty)$ a constant $C_{p,X}$ depending only on X and p exists such that for any two X-valued martingales f and g with tangent martingale difference sequences one has \[\E\|f\|^p \leq…

Let $H$ be a Hilbert space. Using Ball's solution of the "complex plank problem" we prove that the following properties of a sequence $a_n>0$ are equivalent: (1) There is a sequence $x_n \in H$ with $\|x_n\|=a_n$, having 0 as a weak cluster…

We show that if $1<p\neq 2<\infty$, then any isometry of the $p$-convexification of the combinatorial Banach space associated with a hereditary family of finite subsets of $\mathbb{N}$ containing the singletons is given by a signed…

A Banach lattice E is called p-disjointly homogeneous, 1< p< infty, when every sequence of pairwise disjoint normalized elements in E has a subsequence equivalent to the unit vector basis of l_p. Employing methods from interpolation theory,…

Let $P^{k}(n,p) $ be the set of all real polynomial map germs $f = ( f_1 , ..., f_p ) : (\mathbb{R}^{n},0) \rightarrow (\mathbb{R}^{p},0)$ with degree of $f_1 , ...,f_p$ less than or equal to $ k \in \mathbb{N}$. The main result of this…

We prove that, when $2<p<\infty$, in the free Banach lattice generated by $\ell_p$ (respectively by $c_0$), the absolute values of the canonical basis form an $\ell_r$-sequence, where $\frac{1}{r} = \frac{1}{2} + \frac{1}{p}$ (respectively…

In this article, we study some special cases of the problem of classifying polynomials $p:\mathbb{R}^2_+\to (0,\infty)$ for which the net $\{\frac{1}{p(m,n)}\}_{m,n\in \mathbb{Z}_+}$ is a completely monotone net, where $p(x,y)=b(x)+a(x)y$,…

For any set $A$ of natural numbers with positive upper Banach density and any $k\geq 1$, we show the existence of an infinite set $B\subset{\mathbb N}$ and a shift $t\geq0$ such that $A-t$ contains all sums of $m$ distinct elements from $B$…

In this paper we show that every sequence (F_n) of finite dimensional subspaces of a real or complex Banach space with increasing dimensions can be ``refined'' to yield an F.D.D. (G_n), still having increasing dimensions, so that either…

A Banach space $X$ is said to have property (K) if every $w^*$-convergent sequence in $X^*$ admits a convex block subsequence which converges with respect to the Mackey topology. We study the connection of this property with strongly weakly…

In this paper, we discuss the generalized H\"older's inequality in p-summable sequence spaces. In particular, we shall prove sufficient and necessary conditions for generalized H\"older's inequality in those spaces. One of the keys to prove…

A Banach space $X$ is said to have Efremov's property ($\mathcal{E}$) if every element of the weak$^*$-closure of a convex bounded set $C \subseteq X^*$ is the weak$^*$-limit of a sequence in $C$. By assuming the Continuum Hypothesis, we…

A set of sequences is said to converge simultaneously if there exists an infinite subset $H$ of the index set $\omega$ such that all sequences converge when restricted to $H$. We discuss simultaneous convergence of sequences in the same or…

Using an identity arising in the known Banach probability problem on matchboxes, we prove an unexpected congruence for odd prime $p:$ for $1\leq k\leq \frac{p-1}{2},\enskip \sum_{i=1}^{p-2k-1}2^{i-1}\binom{k-1+i}{k}\equiv 0\pmod p.$