We use Tsirelson’s Banach space ([2]) to define an $F_{\sigma }$ P-ideal which refutes a conjecture of Mazur and Kechris (see [12, 9, 8]).

We characterize the approximation property of Banach spaces and their dual spaces by the position of finite rank operators in the space of compact operators. In particular, we show that a Banach space E has the approximation property if and only if for all closed subspaces F of $c_0$, the space ℱ(F,E) of finite rank operators from F to E has the n-intersection property in the corresponding space K(F,E) of compact operators for all n, or equivalently, ℱ(F,E) is an ideal in K(F,E).

Sets of constant width appear as a curiosity in the context of finite-dimensional Euclidean spaces. These sets are convex bodies of such an space with the property that the distance between any two distinct parallel supporting hyperplanes is constant. The easiest example of a set of constant width which is not a ball is the so called Reuleaux triangle in the Euclidean plane. This is the intersection of three closed discs of radius r, whose centers are the vertices of an equilateral triangle of side...

We give new characterizations of Banach spaces not containing ${\ell }_1$ in terms of integral and $p$-dominated polynomials, extending to the polynomial setting a result of Cardassi and more recent results of Rosenthal.

In recent years the study of interpolation of Banach spaces has seen some unexpected interactions with other fields. (...) In this paper I shall discuss some more interactions of interpolation theory with the rest of mathematics, beginning with some joint work with Coifman [CS]. Our basic idea was to look for the methods of interpolation that had interesting PDE's arising as examples.

The interpolation properties of Cesàro sequence and function spaces are investigated. It is shown that $Ce{s}_p\left(I\right)$ is an interpolation space between $Ce{s}_{p₀}\left(I\right)$ and $Ce{s}_{p₁}\left(I\right)$ for 1 < p₀ < p₁ ≤ ∞ and 1/p = (1 - θ)/p₀ + θ/p₁ with 0 < θ < 1, where I = [0,∞) or [0,1]. The same result is true for Cesàro sequence spaces. On the other hand, $Ce{s}_p[0,1]$ is not an interpolation space between Ces₁[0,1] and $Ce{s}_{\infty }[0,1]$.

Let $(M,d)$ be a bounded countable metric space and $c>0$ a constant, such that $d(x,y)+d(y,z)-d(x,z)\ge c$, for any pairwise distinct points $x,y,z$ of $M$. For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of ${\ell }_{\infty }$.

We introduce the definition of $p$-limited completely continuous operators, $1\le p<\infty$. The question of whether a space of operators has the property that every $p$-limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$-limited completely continuous evaluation operators.

It is shown that the weak $L^p$ spaces ${\ell }^{p,\infty },L^{p,\infty }[0,1]$, and $L^{p,\infty }[0,\infty )$ are isomorphic as Banach spaces.