Metric properties of a tensor norm defined by spaces and some characteristics of its associated operator ideals.
Metric Sobolev spaces
We describe an approach to establish a theory of metric Sobolev spaces based on Lipschitz functions and their pointwise Lipschitz constants and the Poincaré inequality.
Metric spaces nonembeddable into Banach spaces with the Radon-Nikodým property and thick families of geodesics
We show that a geodesic metric space which does not admit bilipschitz embeddings into Banach spaces with the Radon-Nikodým property does not necessarily contain a bilipschitz image of a thick family of geodesics. This is done by showing that no thick family of geodesics is Markov convex, and comparing this result with results of Cheeger-Kleiner, Lee-Naor, and Li. The result contrasts with the earlier result of the author that any Banach space without the Radon-Nikodým property contains a bilipschitz...
Metric spaces with the small ball property
A metric space (M,d) is said to have the small ball property (sbp) if for every ε₀ > 0 it is possible to write M as the union of a sequence (B(xₙ,rₙ)) of closed balls such that the rₙ are smaller than ε₀ and lim rₙ = 0. We study permanence properties and examples of sbp. The main results of this paper are the following: 1. Bounded convex closed sets in Banach spaces have sbp only if they are compact. 2. Precisely the finite-dimensional Banach spaces have sbp. (More generally: a complete metric...
Metric unconditionality and Fourier analysis
We investigate several aspects of almost 1-unconditionality. We characterize the metric unconditional approximation property (umap) in terms of “block unconditionality”. Then we focus on translation invariant subspaces and of functions on the circle and express block unconditionality as arithmetical conditions on E. Our work shows that the spaces , p an even integer, have a singular behaviour from the almost isometric point of view: property (umap) does not interpolate between and . These...
Metric version of flatness and Hahn-Banach type theorems for normed modules over sequence algebras
We introduce and study the metric or extreme versions of the notions of a flat and an injective normed module. The relevant definitions, in contrast with the standard known ones, take into account the exact value of the norm of the module. The main result gives a full characterization of extremely flat objects within a certain category of normed modules. As a corollary, some Hahn-Banach type theorems for normed modules are obtained.
Metrically convex functions in normed spaces
Properties of metrically convex functions in normed spaces (of any dimension) are considered. The main result, Theorem 4.2, gives necessary and sufficient conditions for a function to be metrically convex, expressed in terms of the classical convexity theory.
Metrics in the set of partial isometries with finite rank
Let be the set of partial isometries with finite rank of an infinite dimensional Hilbert space . We show that is a smooth submanifold of the Hilbert space of Hilbert-Schmidt operators of and that each connected component is the set , which consists of all partial isometries of rank . Furthermore, is a homogeneous space of , where is the classical Banach-Lie group of unitary operators of , which are Hilbert-Schmidt perturbations of the identity. We introduce two Riemannian metrics...
Metrics in the sphere of a C*-module
Given a unital C*-algebra and a right C*-module over , we consider the problem of finding short smooth curves in the sphere = x ∈ : 〈x, x〉 = 1. Curves in are measured considering the Finsler metric which consists of the norm of at each tangent space of . The initial value problem is solved, for the case when is a von Neumann algebra and is selfdual: for any element x 0 ∈ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ , Z* = −Z and ∥Z∥ ≤ π, such...
Metrics on state spaces.
Metrics on states from actions of compact groups.
Metrizability of precompact sets: an elementary proof.
Proporcionamos una demostración muy corta de un teorema de Cascales-Orihuela que establece que todo conjunto precompacto de un espacio localmente convexo de la clase G (en el sentido de Cascales-Orihuela) es metrizable.
Metrizability of the unit ball of the dual of a quasi-normed cone
We obtain theorems of metrization and quasi-metrization for several topologies of weak* type on the unit ball of the dual of any separable quasi-normed cone. This is done with the help of an appropriate version of the Alaoglu theorem which is also obtained here.
Metrizable [normable] (LF)-spaces and two classical problems in Fréchet [Banach] spaces
Metrization of Compact Convex Sets.
Metrization of uniform lattices
Microfunctions with values in a Clifford algebra 1
Microlocal analysis in the dual of a Colombeau algebra: generalized wave front sets and noncharacteristic regularity.
Microlocal properties of ultradistributions. Composition and kernel type operators.
M-Ideals In Banach Spaces.