Related papers: Differentiability of Banach Spaces via Constructib…

Let S be a dense sub-semigroup of the positive real numbers, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of operators on X over S can be extended to a…

A recent result of T.~Abrahamsen, P.~H\'ajek and S.~Troyanski states that a separable Banach space is almost square if and only if there exists $h\in S_{X^{****}}$ such that $\|x+h\|=\max\{\|x\|,1\}$ for all $x\in X$. The proof passes…

The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset $S$ of a normed…

Let SB be the standard coding for separable Banach spaces as subspaces of $C(\Delta)$. In these notes, we show that if $\mathbb{B} \subset \text{SB}$ is a Borel subset of spaces with separable dual, then the assignment $X \mapsto X^*$ can…

For A a separable unital C*-algebra and M a separable McDuff II_1-factor, we show that the space Hom_w(A,M) of weak approximate unitary equivalence classes of unital *-homomorphisms A \rightarrow M may be considered as a closed, bounded,…

Let $K$ be a Hausdorff space and $C_b(K)$ be the Banach algebra of all complex bounded continuous functions on $K$. We study the G\^{a}teaux and Fr\'echet differentiability of subspaces of $C_b(K)$. Using this, we show that the set of all…

We study classes of Banach spaces where the points of weak$^*$-weak continuity for the identity mapping on the dual unit ball form a weak$^*$-dense and weak$^*$-$G_{\delta}$ set. We also discuss how this property behaves in higher duals of…

In $\mathbb{R}^d$, a closed, convex set has zero Lebesgue measure if and only its interior is empty. More generally, in separable, reflexive Banach spaces, closed and convex sets are Haar null if and only if their interior is empty. We…

We prove that, given any covering of any separable infinite-dimensional uniformly rotund and uniformly smooth Banach space $X$ by closed balls each of positive radius, some point exists in $X$ which belongs to infinitely many balls.

Let X be a separable real Banach space having a k-times continuously Fr\'{e}chet differentiable (i.e. C^k-smooth) norm where k=1,...,\infty. We show that any equivalent norm on X can be approximated uniformly on bounded sets by C^k-smooth…

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…

We construct a differentiable locally Lipschitz function $f$ in $\mathbb{R}^{N}$ with the property that for every convex body $K\subset \mathbb{R}^N$ there exists $\bar x \in \mathbb{R}^N$ such that $K$ coincides with the set $\partial_L…

A separable Banach space X contains $\ell_1$ isomorphically if and only if X has a bounded wc_0^*-stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded wc_0^*-biorthogonal…

A subset $A$ of a Banach space is called Banach-Saks when every sequence in $A$ has a Ces{\`a}ro convergent subsequence. Our interest here focusses on the following problem: is the convex hull of a Banach-Saks set again Banach-Saks? By…

We show that if a Banach space X has the weak fixed point property for nonexpansive mappings and Y has the generalized Gossez-Lami Dozo property or is uniformly convex in every direction, then the direct sum of X and Y with a strictly…

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…

We demonstrate that a reproducing kernel Hilbert or Banach space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel measurable feature map.

We show that there exists a separable reflexive Banach space into which every separable uniformly convex Banach space isomorphically embeds. This solves a problem of J. Bourgain. We also give intrinsic characterizations of separable…

It is known that a Banach space has the strong diameter 2 property (i.e. every convex combination of slices of the unit ball has diameter 2) if and only if the norm on its dual space is octahedral (a notion introduced by Godefroy and…

A Banach space $X$ is called subprojective if any of its infinite dimensional subspaces $Y$ contains a further infinite dimensional subspace complemented in $X$. This paper is devoted to systematic study of subprojectivity. We examine the…