Exemples d'espaces de Banach ayant la propriété de projection uniforme
We consider a compact linear map T acting between Banach spaces both of which are uniformly convex and uniformly smooth; it is supposed that T has trivial kernel and range dense in the target space. It is shown that if the Gelfand numbers of T decay sufficiently quickly, then the action of T is given by a series with calculable coefficients. This provides a Banach space version of the well-known Hilbert space result of E. Schmidt.
Given a subset A of a topological space X, a locally convex space Y, and a family ℂ of subsets of Y we study the problem of the existence of a linear ℂ-extender , which is a linear operator extending bounded continuous functions f: A → C ⊂ Y, C ∈ ℂ, to bounded continuous functions f̅ = u(f): X → C ⊂ Y. Two necessary conditions for the existence of such an extender are found in terms of a topological game, which is a modification of the classical strong Choquet game. The results obtained allow us...
We show that a Banach space X is an ℒ₁-space (respectively, an -space) if and only if it has the lifting (respectively, the extension) property for polynomials which are weakly continuous on bounded sets. We also prove that X is an ℒ₁-space if and only if the space of m-homogeneous scalar-valued polynomials on X which are weakly continuous on bounded sets is an -space.
These notes deal with the extension of multilinear operators on Banach spaces. The organization of the paper is as follows. In the first section we study the extension of the product on a Banach algebra to the bidual and some related structures including modules and derivations. Tha approach is elementary and uses the classical Arens' technique. Actually most of the results of section 1 can be easily derived from section 2. In section 2 we consider the problem of extending multilinear forms on a...
It is proved that every operator from a weak*-closed subspace of into a space C(K) of continuous functions on a compact Hausdorff space K can be extended to an operator from to C(K).
It follows from our earlier results [Israel J. Math., to appear] that in the Gurariy space G every finite-dimensional smooth subspace is contained in a bigger smooth subspace. We show that this property does not characterise the Gurariy space among Lindenstrauss spaces and we provide various examples to show that C(K) spaces do not have this property.
We study extension operators between spaces of continuous functions on the spaces of subsets of X of cardinality at most n. As an application, we show that if is the unit ball of a nonseparable Hilbert space H equipped with the weak topology, then, for any 0 < λ < μ, there is no extension operator .
We present two types of representation theorems: one for linear continuous operators on space of Banach valued regulated functions of several real variables and the other for bilinear continuous operators on cartesian products of spaces of regulated functions of a real variable taking values on Banach spaces. We use generalizations of the notions of functions of bounded variation in the sense of Vitali and Fréchet and the Riemann-Stieltjes-Dushnik or interior integral. A few applications using geometry...
Soit un sous-espace fermé d’un espace de Banach ordonné ; ce travail propose des conditions nécessaires et suffisantes pour qu’il existe , tel que toute forme linéaire positive et continue sur admette une extension linéaire positive et continue sur , vérifiant . On termine par l’exemple d’un couple ne possédant pas la propriété précédente bien que toute forme linéaire positive continue sur se prolonge en une forme linéaire du même type en .