Related papers: A 1-separably injective space that does not contai…

We consider the closed subspace of $\ell_\infty$ generated by $c_0$ and the characteristic functions of elements of an uncountable, almost disjoint family $\mathcal A$ of infinite subsets of $\mathbb N$. This Banach space has the form…

Assuming $\mathfrak p=\mathfrak c$, we show that for every Eberlein compact space $L$ of weight $\mathfrak c$ there exists a short exact sequence $0\to c_0\to X\to C(L)\to 0$, where the Banach space $X$ is not isomorphic to a $C(K)$-space.

This article is a continuation of a paper of the first author \cite{F} about complex structures on real Banach spaces. We define a notion of even infinite dimensional real Banach space, and prove that there exist even spaces, including HI…

This paper is inspired by the paper of Leonetti, Russo and Somaglia [\textit{Dense lineability and spaceability in certain subsets of $\ell_\infty$.} Bull. London Math. Soc., 55: 2283--2303 (2023)] and the lineability problems raised…

We consider the problem of determining the complexity of the uniform homeomorphism relation between separable Banach spaces in the Borel reducibility hierarchy of analytic equivalence relations. We prove that the complete $K_{\sigma}$…

We prove that Banach spaces $\ell_1\oplus_2\mathbb{R}$ and $X\oplus_\infty Y$, with strictly convex $X$ and $Y$, have plastic unit balls (we call a metric space plastic if every non-expansive bijection from this space onto itself is an…

In 1970 Haskell Rosenthal proved that if $X$ is a Banach space, $\Gamma$ is an infinite index set, and $T:\ell_\infty(\Gamma)\to X$ is a bounded linear operator such that $\inf_{\gamma\in\Gamma}\|T(e_\gamma)\|>0$ then $T$ acts as an…

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 for a $\sigma $-ideal $\ci$ with a Steinhaus property defined on Banach space, if two non-homeomorphic Banach with the same cardinality of the Hamel basis then there is a $\ci$ nonmeasurable subset as image by any isomorphism…

We construct a Banach space that does not contain any infinite unconditional basic sequence.

We prove that hereditarily Lindel\"of space which is $F_{\sigma\delta}$ in some compactification is absolutely $F_{\sigma\delta}$. In particular, this implies that any separable Banach space is absolutely $F_{\sigma\delta}$ when equipped…

We show that for "most" compact non metrizable spaces, the unit ball of the Banach space C(K) contains an uncountable 2-equilateral set. We also give examples of compact non metrizable spaces K such that the minimum cardinality of a maximal…

We study certain Banach spaces that are added in the extension by one Cohen real. Specifically, we show that adding just one Cohen real to any model adds a Banach space of density $\aleph_1$ which does not embed into any such space in the…

This paper considers explicit constructions of Auerbach bases in separable Banach spaces. Answering the question of A. Pe{\l}czy{\'n}ski, we prove by construction the existence of Auerbach basis in arbitrary subspace of $c_0$ of finite…

We examine the phenomenon when surjective algebra homomorphisms between algebras of operators on Banach spaces are automatically injective. In the first part of the paper we shall show that for certain Banach spaces $X$ the following…

James Tree Space ($\mathcal{JT}$), introduced by R. James, is the first Banach space constructed having non-separable conjugate and not containing $\ell^1$. James actually proved that every infinite dimensional subspace of $\mathcal{JT}$…

The present paper contributes to the ongoing programme of quantification of isomorphic Banach space theory focusing on Pe{\l}czy\'nski's classical work on dual Banach spaces containing $L_{1}$ ($=L_{1}[0,1]$) and the Hagler--Stegall…

While the classic separable quotient problem remains open, we survey general results related to this problem and examine the existence of a particular infinitedimensional separable quotient in some Banach spaces of vector-valued functions,…

For a Banach space $X$ its subset $Y\subseteq X$ is called overcomplete if $|Y|=dens(X)$ and $Z$ is linearly dense in $X$ for every $Z\subseteq Y$ with $|Z|=|Y|$. In the context of nonseparable Banach spaces this notion was introduced…

In this paper, we study the descriptive complexity of some inevitable classes of Banach spaces. Precisely, as shown in [Go], every Banach space either contains a hereditarily indecomposable subspace or an unconditional basis, and, as shown…