Related papers: Coordinate systems in Banach spaces and lattices

We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use $\ell_{\Psi}$-Hilbertian and $\infty$-Hilbertian…

The distinction between conditional, unconditional, and absolute convergence in infinite-dimensional spaces has fundamental implications for computational algorithms. While these concepts coincide in finite dimensions, the Dvoretzky-Rogers…

For every filter $\mathcal F$ on $\mathbb N$, we introduce and study corresponding uniform $\mathcal F$-boundedness principles for locally convex topological vector spaces. These principles generalise the classical uniform boundedness…

We provide a new extension of Pitt's theorem for compact operators between quasi-Banach lattices, which permits to describe unconditional bases of finite direct sums of Banach spaces $\mathbb{X}_{1}\oplus\dots\oplus\mathbb{X}_{n}$ as direct…

We study the notions of acs, luacs and uacs Banach spaces which were introduced by V. Kadets et al. in 2000 and form common generalisations of the usual rotundity and smoothness properties of Banach spaces. In particular, we are interested…

We characterize the solution of a broad class of convex optimization problems that address the reconstruction of a function from a finite number of linear measurements. The underlying hypothesis is that the solution is decomposable as a…

We give a detailed proof D. Handelman's theorem stating (in the context of an order unit normed space) that a monotone sigma-complete order unit normed space is a Banach space.

The purpose of this note is to show that, if $\mcB$ is a uniformly convex Banach, then the dual space $\mcB'$ has a "Hilbert space representation" (defined in the paper), that makes $\mcB$ much closer to a Hilbert space then previously…

In this paper we prove coincidence results concerning spaces of absolutely summing multilinear mappings between Banach spaces. The nature of these results arises from two distinct approaches: the coincidence of two \textit{a priori}…

The concept of uniform convexity of a Banach space was generalized to linear operators between Banach spaces and studied by Beauzamy [1976]. Under this generalization, a Banach space X is uniformly convex if and only if its identity map I_X…

We show examples of compact linear operators between Banach spaces which cannot be approximated by norm attaining operators. This is the negative answer to an open question posed in the 1970's. Actually, any strictly convex Banach space…

Reordering the terms of a series is a useful mathematical device, and much is known about when it can be done without affecting the convergence or the sum of the series. For example, if a series of real numbers absolutely converges, we can…

In the present note, the Banach contraction principle is proved in complete modular spaces via an order theoretic approach.

The aim of this paper is to present a tool used to show that certain Banach spaces can be endowed with $C^k$ smooth equivalent norms. The hypothesis uses particular countable decompositions of certain subsets of $B_{X^*}$, namely…

We show that the centre of a Dedekind complete complex Banach lattice is a commutative $\mathrm{C}^\ast$-algebra in the order unit norm. This implies that the order unit norm and the operator norm coincide. As an application of the latter,…

It is a longstanding problem whether every contractible Banach algebra is necessarily finite-dimensional. In this note, we confirm this for Banach algebras acting on Banach spaces with the uniform approximation property. This generalizes a…

We present a Banach space $\mathfrak X$ with a Schauder basis of length $\omega\_1$ which is saturated by copies of $c\_0$ and such that for every closed decomposition of a closed subspace $X=X\_0\oplus X\_1$, either $X\_0$ or $X\_1$ has to…

This paper deals with the problem of when, given a collection $\mathcal C$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space $Z$ with a Schauder basis so that every element in…

In this paper, we introduce, for a separable Banach spacea new notion of besselian paires and of besselian Schauder frames for which we prove for some fundamental results.

We study the optimal upper $X_U$ and lower $X_L$ sequence spaces that can be assigned to each Banach lattice $X$. These spaces are symmetric, have the Fatou property and the unit vector basis has in these spaces very special properties.…