Related papers: A solution of one J. Bourgain's problem

For every Banach space $Z$ with a shrinking unconditional basis satisfying upper $p$-estimates for some $p > 1$, an isomorphically polyhedral Banach space is constructed having an unconditional basis and admitting a quotient isomorphic to…

We set up a descriptive set-theoretic framework to study Lipschitz-free spaces and use the reduction argument of Bossard to prove several results. We prove two universality results: if a separable Banach space is isomorphically universal…

Mazur's separable quotient problem, open since 1932, asks whether every infinite-dimensional Banach space admits an infinite-dimensional separable quotient. We prove that any $\mathscr{L}_\infty$-space $Y$ containing a subspace $X$ such…

We answer in the affirmative the surprisingly difficult questions: If a complex Banach space possesses a real predual X, then is X a complex Banach space? If a complex Banach space possesses a real predual, then does it have a complex…

The Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient space, has remained unsolved for 85 years, but has been answered in the affirmative for special cases such as…

We show that, if $X$ is a closed subspace of a Banach space $E$ and $Z$ is a closed subspace of $E^*$ such that $Z$ is norming for $X$ and $X$ is total over $Z$ (as well as $X$ is norming for $Z$ and $Z$ is total over $X$), then $X$ and the…

We give in this short report a very simple proof that arbitrary random variable with Borelian distribution in separable Banach space belongs with probability one to a pre-image of some linear compact non-random operator.

In this paper we present some new results on the existence of solutions of generalized variational inequalities in real reflexive Banach spaces with Fr\'echet differentiable norms. Moreover, we also give some theorems about the structure of…

A final-state observability result in the Banach space setting for non-autonomous observation problems is obtained that covers and extends all previously known results in this context, while providing a streamlined proof that follows the…

We extend the well known criteria of reflexivity of Banach lattices due to Lozanovsky and Lotz to the class of finitely generated Banach $C(K)$- modules. Namely we prove that a finitely generated Banach $C(K)$-module is reflexive if and…

The aim of this note is to prove that, given two superreflexive Banach spaces $X$ and $Y$, then $X\widehat{\otimes}_\pi Y$ is superreflexive if and only if either $X$ or $Y$ is finite-dimensional. In a similar way, we prove that…

A reflexive hereditarily indecomposable Banach space $\mathfrak{X}_{_{^\text{ISP}}}$ is presented, such that for every $Y$ infinite dimensional closed subspace of $\mathfrak{X}_{_{^\text{ISP}}}$ and every bounded linear operator…

This note has two objectives. The first objective is show that, even if a separable Banach space does not have a Schauder basis (S-basis), there always exists Hilbert spaces $\mcH_1$ and $\mcH_2$, such that $\mcH_1$ is a continuous dense…

In this note the following is proved. Separable L-embedded spaces - that is separable Banach spaces which are complemented in their biduals such that the norm between the two complementary subspaces is additive - have property (X) which, by…

Following results of Bourgain and Gorelik we show that the spaces $\ell_p$, $1<p<\infty$, as well as some related spaces have the following uniqueness property: If $X$ is a Banach space uniformly homeomorphic to one of these spaces then it…

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a…

Given a category of objects, it is both useful and important to know if all the objects in the category may be realised as sub-objects -- via morphisms in the given category -- of a single object in that category enjoying some nice…

We show that the problem whether every $1$-separably injective Banach space contains an isomorphic copy of $\ell_\infty$ is undecidable. Namely, unlike under the continuum hypothesis, assuming Martin's axiom and the negation of the…

For any closed subset $F$ of $[1,\infty]$ which is either finite or consists of the elements of an increasing sequence and its limit, a reflexive Banach space $X$ with a 1-unconditional basis is constructed so that in each block subspace…

In this paper, we prove the existence and uniqueness of the solution $f(z,t)$ of the Loewner PDE with normalization $Df(0,t)=e^{tA}$, where $A\in L(X,X)$ is such that $k_+(A)<2m(A)$, on the unit ball of a separable reflexive complex Banach…