Let $ℳ$ be the Banach space of real measures on a $\sigma$-ring $\mathbf{R}$, let ${ℳ}^{\text{'}}$ be its dual, let $E$ be a quasi-complete locally convex space, let $E^{\text{'}}$ be its dual, and let $\mu$ be an $E$-valued measure on $\mathbf{R}$. If is shown that for any $\theta \in {ℳ}^{\text{'}}$ there exists an element $\int \theta d\mu$ of $E$ such that $⟨x^{\text{'}}\circ \mu ,\theta ⟩=⟨\int \theta d\mu ,x^{\text{'}}⟩$ for any $x^{\text{'}}\in E^{\text{'}}$ and that the map$$\theta \to \int \theta d\mu :{ℳ}^{\text{'}}\to E$$is order continuous. It follows that the closed convex hull of $\mu \left(\mathbf{R}\right)$ is weakly compact.

Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). For a real Banach space $(X,|\left|·\right|{|}_X)$ 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 ${\left|\right|f\left(·\right)\left|\right|}_X$ belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let $D_u(={f\in E\left(X\right):\left|\right|f\left(·\right)\left|\right|}_X\le u)$ stand for the order interval in E(X). For a real Banach space $(Y,|\left|·\right|{|}_Y)$ a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈ E⁺ the set...

For $U$ a balanced open subset of a Fréchet space $E$ and $F$ a dual-Banach space we introduce the topology ${\tau }_{\gamma }$ on the space $\mathscr{H}(U,F)$ of holomorphic functions from $U$ into $F$. This topology allows us to construct a predual for $(\mathscr{H}(U,F),{\tau }_{\delta })$ 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 $T$, on désigne par $M_{\beta }\left(T\right)$ l’espace des mesures de Radon sur le compactifié de Stone-Cech $\beta T$ de $T$ et par $M_{\sigma }\left(T\right)$ son sous-espace formé des mesures $\sigma$-régulières au sens de Varadarajan. On décrit alors sur ces deux espaces des topologies ${\mathbf{T}}_p$, $1\le p\le +\infty$, qui possèdent des propriétés curieuses parmi lesquelles il convient de citer la suivante : pour $1\<p\le +\infty$ et pour tout $T$ non pseudocompact, l’espace $\left(M_{\sigma }\right(T),{\mathbf{T}}_p)$ 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.