On the strict convexity of the polar operator
Let be the Banach space of real measures on a -ring , let be its dual, let be a quasi-complete locally convex space, let be its dual, and let be an -valued measure on . If is shown that for any there exists an element of such that for any and that the mapis order continuous. It follows that the closed convex hull of is weakly compact.
Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). For a real Banach space let E(X) be a subspace of the space L⁰(X) of μ-equivalence classes of strongly Σ-measurable functions f: Ω → X and consisting of all those f ∈ L⁰(X) for which the scalar function belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let stand for the order interval in E(X). For a real Banach space a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈ E⁺ the set...
For a balanced open subset of a Fréchet space and a dual-Banach space we introduce the topology on the space of holomorphic functions from into . This topology allows us to construct a predual for which in turn allows us to investigate the topological structure of spaces of vector-valued holomorphic functions. In particular, we are able to give necessary and sufficient conditions for the equivalence and compatibility of various topologies on spaces of vector-valued holomorphic functions....
Pour tout compact complètement régulier , on désigne par l’espace des mesures de Radon sur le compactifié de Stone-Cech de et par son sous-espace formé des mesures -régulières au sens de Varadarajan. On décrit alors sur ces deux espaces des topologies , , qui possèdent des propriétés curieuses parmi lesquelles il convient de citer la suivante : pour et pour tout non pseudocompact, l’espace est non quasi-complet mais ses précompacts sont relativement compacts. Ce résultat permet...
Over the past few years there has been considerable progress in the structural understanding of special Colombeau algebras. We present some of the main trends in this development: non-smooth differential geometry, locally convex theory of modules over the ring of generalized numbers, and algebraic aspects of Colombeau theory. Some open problems are given and directions of further research are outlined.