Soient , des espaces de Banach , des espaces d’Orlicz, on définit les applications sommantes de dans . On montre que de telles applications sont radonifiantes de dans .On donne une factorisation caractéristique des applications sommantes.
Let ϰ be a positive, continuous, submultiplicative function on such that for some ω ∈ ℝ, α ∈ and . For every λ ∈ (ω,∞) let for . Let be the space of functions Lebesgue integrable on with weight , and let E be a Banach space. Consider the map . Theorem 5.1 of the present paper characterizes the range of the linear map defined on , generalizing a result established by B. Hennig and F. Neubrander for . If ϰ ≡ 1 and E =ℝ then Theorem 5.1 reduces to D. V. Widder’s characterization...
We prove that several results of Talagrand proved for the Pettis integral also hold for the Kurzweil-Henstock-Pettis integral. In particular the Kurzweil-Henstock-Pettis integrability can be characterized by cores of the functions and by properties of suitable operators defined by integrands.
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...
A characterization is given of those Banach-space-valued vector measures m with finite variation whose associated integration operator Iₘ: f ↦ ∫fdm is compact as a linear map from L¹(m) into the Banach space. Moreover, in every infinite-dimensional Banach space there exist nontrivial vector measures m (with finite variation) such that Iₘ is compact, and other m (still with finite variation) such that Iₘ is not compact. If m has infinite variation, then Iₘ is never compact.