On Orlicz spaces of vector-valued functions. (Spanish: Sobre los espacios de Orlicz de funciones vectoriales)

Thumbnail Image
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Google Scholar
Research Projects
Organizational Units
Journal Issue
Let (S,Σ,μ) be a finite measure space, X a Banach space and Φ a Young function with complementary function Ψ. There is a natural duality between the Orlicz spaces LΦ(X) and LΨ(X∗), given by (f,g)↦∫⟨f,g⟩dμ. Assume that X satisfies the Radon-Nikodým property. One of the main results obtained in this paper is the following: K⊂LΦ(X) is σ(LΦ(X),LΨ(X∗)) relatively sequentially compact if and only if the following conditions are satisfied: (i) K is norm-bounded, (ii) the set K(A)={∫Afdμ:f∈K} is relatively weakly compact in X for every A∈Σ, and (iii) limμ(A)→0sup{∫A⟨f,g⟩dμ:f∈K}=0 for every g∈LΨ(X∗).
Ando, T. (1962). Weakly compact sets in Orlicz-spaces. Canad. J. of Math., 14, 170-176. BATT, J. (1974). On weak compactness in spaces of vector- values measures and Bochner-integrable functions in connection with the Radon-Nikodym property of Banach spaces. Rev. Roum. Math. Pures et Appl., XIX, 285-304. BOMBAL, F. (1980). Sobre el espacio L p (u, X). Rev. Acad. Ciencias de Madrid, LXXIV, 131-135. DIESTEL, J. AND UHL, J. (1977). Vector Measures. Math. Surveys, nº. 15. Amer. Math. Soc., Providencc, RI. DUNFORD, N. and SCHWARTZ, J.T. (1966). Linear operators, Part I. Interscience Pub. New York. KUFNER et ALT. (1977). Function spaces. Noordhoff Int. Leyden LuxEMBURG, W.A.J. (1955). Banach fuuntion spaces. Tesis, Delft. UHL, J.J. (1969). Applications of Radon-Nikodym theoremstlo martingale convergence. Trans. of the Amer. Math. Soc., 145, 271-285. ZAANEN, A.C. (1953). Linear Analysis. North-Holland Pub. CO. Amsterdam