Related papers: Local symmetry and smoothness in the space of vect…

We study symmetric points with respect to $(\rho_+)$-orthogonality, $(\rho_{-})$-orthogonality and $\rho$-orthogonality in the space $C(K, \mathbb{X}),$ where $K$ is a perfectly normal, compact space and $ \mathbb X$ is a Banach space. We…

We identify the smooth points of $L^1(\mu,X)$, and provide some necessary and sufficient conditions for left and right symmetry of points with respect to Birkhoff-James orthogonality in $L^p(\mu,X), 1\leq p<\infty$, where $\mu$ is any…

We study left symmetric and right symmetric elements in the space $\ell_{\infty}(K, \mathbb{X}) $ of bounded functions from a non-empty set $K$ to a Banach space $\mathbb{X}.$ We prove that a non-zero element $ f \in\ell_{\infty}(K,…

We completely characterize the left-symmetric points, the right-symmetric points, and, the symmetric points in the sense of Birkhoff-James, in a Banach space. We obtain a complete characterization of the left-symmetric (right-symmetric)…

Let $C_0(K, X)$ denote the space of all continuous $X$-valued functions defined on the locally compact Hausdorff space $K$ which vanish at infinity, provided with the supremum norm. If $X$ is the scalar field, we denote $C_0(K, X)$ by…

Suppose $X$ and $Y$ are Banach spaces, $K$ is a compact Hausdorff space, $\Sigma$ is the $\sigma$-algebra of Borel subsets of $K$, $C(K,X)$ is the Banach space of all continuous $X$-valued functions (with the supremum norm), and…

The paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $\Delta$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_0$ which is isomorphic to a closed subspace of a…

A Banach symmetric space in the sense of O. Loos is a smooth Banach manifold $M$ endowed with a multiplication map $\mu\colon M \times M \to M$ such that each left multiplication map $\mu_x := \mu(x,\cdot)$ (with $x \in M$) is an involutive…

Let $X$ be a compact Hausdorff space and $A$ a Banach algebra. We investigate amenability properties of the algebra $C(X,A)$ of all $A$-valued continuous functions. We show that $C(X,A)$ has a bounded approximate diagonal if and only if $A$…

These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach-Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach…

Let X be a compact Hausdorff space and M a metric space. E_0(X,M) is the set of f in C(X,M) such that there is a dense set of points x in X with f constant on some neighborhood of x. We describe some general classes of X for which E_0(X,M)…

Let $E$ be a Banach space and $\X$ be the closed unit ball of the dual space $E^*$. For a compact set $K$ in $E$, we prove that $K$ is polynomially convex in $E$ if and only if there exist a unital commutative Banach algebra $A$ and a…

We characterize the Banach spaces X such that Ext(X, C(K))=0 for every compact space.

Let $A$ be a complex Banach space with a norm $\|f\|=\|f\|_X+\|d(f)\|_Y$ for $f\in A$, where $d$ is a complex linear map from $A$ onto a Banach space $B$, and $\|\cdot\|_K$ represents the supremum norm on a compact Hausdorff space $K$. In…

We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces $L_X^p$, where $X$ is a Banach space and $1\le p<\infty$, and extend the result to vector-valued Banach function spaces…

In this paper, we discuss the stability of (restricted) Chebyshev centers in few function spaces. For an extremally disconnected compact Hausdorff space $K$ and a finite dimensional Banach space $X$, we prove the existence of Chebyshev…

A classical result of Cembranos and Freniche states that the C(K, X) spaces contains a complemented copy of c_0 whenever K is an infinite compact Hausdorff space and X is an infinite dimensional Banach space. This paper takes this result as…

We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be…

Let $K_1$, $K_2$ be compact Hausdorff spaces and $E_1, E_2$ be Banach spaces not containing a copy of $c_0$. We establish lower estimates of the Banach-Mazur distance between the spaces of continuous functions $\mathcal{C}(K_1, E_1)$ and…

We deal with isomorphic Banach-Stone type theorems for closed subspaces of vector-valued continuous functions. Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. For $i=1,2$, let $E_i$ be a reflexive Banach space over $\mathbb{F}$ with a certain…