A bounded closed convex set K in a Banach space X is said to have quasi-normal structure if each bounded closed convex subset H of K for which diam(H) > 0 contains a point u for which ∥u-x∥ < diam(H) for each x ∈ H. It is shown that if the convex sets on the unit sphere in X satisfy this condition (which is much weaker than the assumption that convex sets on the unit sphere are separable), then relative to various weak topologies, the unit ball in X is compact whenever it is countably compact....

The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for $L²\left({ℝ}^d\right)$, and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin, modulation and Bargmann-Fock spaces.

In this paper we give a characterization of the relatively compact subsets of the so-called approximation spaces. We treat some applications: (1) we obtain some convergence results in such spaces, and (2) we establish a condition for relative compactness of a set lying in a Besov space.

We study compactness and related topological properties in the space L¹(m) of a Banach space valued measure m when the natural topologies associated to convergence of vector valued integrals are considered. The resulting topological spaces are shown to be angelic and the relationship of compactness and equi-integrability is explored. A natural norming subset of the dual unit ball of L¹(m) appears in our discussion and we study when it is a boundary. The (almost) complete continuity of the integration...

We prove an abstract comparison principle which translates gaussian cotype into Rademacher cotype conditions and vice versa. More precisely, let 2 < q < ∞ and T: C(K) → F a continuous linear operator. (1) T is of gaussian cotype q if and only if ${\left({\sum }_k{(\left(\parallel T{x}_k{\parallel }_F\right)/\left(\surd log(k+1)\right))}^q\right)}^{1/q}\le c\parallel {\sum }_k{\varepsilon }_k{x}_k{\parallel }_{L_2\left(C\left(K\right)\right)}$, for all sequences ${\left(x_k\right)}_{k\in \mathbb{N}}\subset C\left(K\right)$ with ${\left(\parallel T{x}_k\parallel \right)}_{k=1}^n$ decreasing. (2) T is of Rademacher cotype q if and only if $({\sum }_k{\left(\parallel T{x}_k{\parallel }_F\surd \left({\left(log(k+1)\right)}^q\right)\right)}^{1/q}\le c\parallel {\sum }_k{g}_k{x}_k{\parallel }_{L_2\left(C\left(K\right)\right)}$, for all sequences ${\left(x_k\right)}_{k\in \mathbb{N}}\subset C\left(K\right)$ with ${\left(\parallel T{x}_k\parallel \right)}_{k=1}^n$ decreasing. Our method allows a restriction to a fixed number of vectors and complements the corresponding results of Talagrand.

Let A = (aij) ∊ Mn(ℝ) be an n by n symmetric stochastic matrix. For p ∊ [1, ∞) and a metric space (X, dX), let γ(A, dpx) be the infimum over those γ ∊ (0,∞] for which every x1, . . . , xn ∊ X satisfy [...] Thus γ (A, dpx) measures the magnitude of the nonlinear spectral gap of the matrix A with respect to the kernel dpX : X × X →[0,∞). We study pairs of metric spaces (X, dX) and (Y, dY ) for which there exists Ψ: (0,∞)→(0,∞) such that γ (A, dpX) ≤Ψ (A, dpY ) for every symmetric stochastic A ∊ Mn(ℝ)...

For a locally convex space E we prove that the space of n-symmetric tensors is complemented in the space of (n+1)-symmetric tensors endowed with the projective topology. Applications and related results are also given.

Does a given Banach space have any non-trivial complemented subspaces? Usually, the answer is: yes, quite a lot. Sometimes the answer is: no, none at all.

En esta nota consideramos una clase de espacios topológicos de Hausdorff localmente compactos (Ω) con la propiedad de que el espacio de Banach C0(Ω) de todas las funciones continuas con valores escalares definidas en Ω que se anulan en el infinito, equipado con la norma supremo, contiene una copia de C0 norma-uno complementada, mientras que C (βΩ) contiene una copia de l∞ linealmente isométrica.

Let [Lambda] be a barrelled perfect (in the sense of J. Dieudonné) Köthe space of measurable functions defined on an atomless finite Radon measure space. Let X be a Banach space containing a copy of c0, then the space [Lambda(X)] of [Lambda]-Bochner integrable functions contains a complemented copy of c0.

We give sufficient conditions on Banach spaces $X$ and $Y$ so that their projective tensor product $X{\otimes }_{\pi }Y$, their injective tensor product $X{\otimes }_{ϵ}Y$, or the dual ${\left(X{\otimes }_{\pi }Y\right)}^{*}$ contain complemented copies of ${\ell }_p$.

We prove that the direct sum and the product of countably many copies of L1[0, 1] are primary locally convex spaces. We also give some related results.