Related papers: Lindelof type of generalization of separability in…

We construct infinitely differentiable norms and partitions of unity for a class of Banach spaces which includes all spaces $\C(K)$ with $K$ a countable compact space, and all spaces $\C_0[0,\Omega )$ with $\Omega $ an ordinal.

The famous Rosenthal-Lacey theorem asserts that for each infinite compact set $K$ the Banach space $C(K)$ admits a quotient which is either a copy of $c$ or $\ell_{2}$. What is the case when the uniform topology of $C(K)$ is replaced by the…

We show that any non-zero Banach space with a separable dual contains a totally disconnected, closed and bounded subset S of Hausdorff dimension 1 such that every Lipschitz function on the space is Fr\'echet differentiable somewhere in S.

Given an uncountable subset $\mathcal Y$ of a nonseparable Banach space, is there an uncountable $\mathcal Z\subseteq \mathcal Y$ such that the distances between any two distinct points of $\mathcal Z$ are more or less the same? If an…

We study the class of compact spaces that appear as structure spaces of separable Banach lattices. In other words, we analyze what $C(K)$ spaces appear as principal ideals of separable Banach lattices. Among other things, it is shown that…

We extend the well-known characterizations of convergence in the spaces $l_p$ ($1\le p<\infty$) of $p$-summable sequence and $c_0$ of vanishing sequences to a general characterization of convergence in a Banach space with a Schauder basis…

A separable Banach space $X$ is said to be finitely determined if for each separable space $Y$ such that $X$ is finitely representable (f.r.) in $Y$ and $Y$ is f.r. in $X$ then $Y$ is isometric to $X$. We provide a direct proof (without…

Let X be a complex Banach space, in this work we characterize the property of Frechet differentiability for the dual space of X. In the following, we show that if the dual space of X is Gateaux differentiable, then the dual space of Lp(X)…

It is shown that a separable Banach space $X$ can be given an equivalent norm $|\!|\!|\cdot |\!|\!|$ with the following properties:\quad If $(x_n)\subseteq X$ is relatively weakly compact and $\lim_{m\to\infty} \lim_{n\to\infty}\break…

We prove that if X is an infinite-dimensional Banach space with C^p smooth partitions of unity, then X and X\K are C^p diffeomorphic, for every weakly compact subset K of X.

We give a characterisation of the separable Banach spaces with the Daugavet property which is applied to study the Daugavet property in the projective tensor product of an $L$-embedded space with another non-zero Banach space. The former…

If a separable Banach space contains an isometric copy of every separable reflexive Fr\'echet smooth Banach space, then it contains an isometric copy of every separable Banach space. The same conclusion holds if we consider separable Banach…

Generalizations of the theorems of Eberlein and Grothendieck on the precompactness of subsets of function spaces are considered: if $X$ is a countably compact space and $C_p(X)$ is a space of continuous functions in the pointwise topology…

We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $\ell_1$. We present many examples and several properties of this class. We give some…

We consider the cardinal invariant CG(X) of the minimal number of weakly compact subsets which generate a Banach space X. We study the behavior of this index when passing to subspaces, its relation with the Lindelof number in the weak…

In this paper, we study spaceability of subsets of generalized Orlicz and Lebesgue spaces associated to Banach function space. Also, we give some sufficient conditions for spaceability of subsets of a general Banach space which improves an…

It is shown that every separable reflexive Banach space is a quotient of a reflexive Hereditarily Indecomposable space, which yields that every separable reflexive Banach is isomorphic to a subspace of a reflexive Indecomposable space.…

A display of a topological group G on a Banach space X is a topological isomorphism of G with the isometry group Isom(X,||.||) for some equivalent norm ||.|| on X, where the latter group is equipped with the strong operator topology.…

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,…

We characterize those classes $\mathcal{C}$ of separable Banach spaces for which there exists a separable Banach space $Y$ not containing $\ell_1$ and such that every space in the class $\mathcal{C}$ is a quotient of $Y$.