Duality and some topological properties of vector-valued function spaces

• Autorzy: Feledziak K.
• Języki publikacji: EN

Abstrakty

EN

Let E be an ideal of L^0 over σ-finite measure space (Ω,Σ μ) and let (X, || X) be a real Banach space. Let E(X) be a subspace of the space L^0(X) of μ-equivalence classes of all strongly Σ-measurable functions f : Ω → X and consisting of all those f ε L^0(X), for which the scalar function [...] belongs to E. Let E be equipped with a Hausdorff locally convex-solid topology ξ and let ξ stand for the topology on E(X) associated with ξ. We examine the relationship between the properties of the space (E(X), ξ) and the properties of both the spaces (E, ξ) and (X, ||X). In particular, it is proved that E(X) (embedded in a natural way) is an order closed ideal of its bidual iff E is an order closed ideal of its bidual and X is reflexive. As an application, we obtain that E(X) is perfect iff E is perfect and X is reflexive.

Słowa kluczowe

EN

vector-valued function spaces; locally solid topologies; KB-spaces; Levy topologies; Lebesgue topologies; order dual; order continuous dual; perfectness

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2008
• Tom: Vol. 48, [Z] 1
• Strony: 23--43
• Opis fizyczny: bibliogr. 21 poz.

Twórcy

autor

Feledziak K.

• Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra Szafrana 4A, 65-516 Zielona Góra, Poland,

Bibliografia

• [1] C. D. Aliprantis, O. Burkinshaw, Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs, Volume 105, American Mathematical Society 2003.
• [2] C. D. Aliprantis, O. Burkinshaw, Positive Operators, Academic Press, New York, 1985.
• [3] F. Bombal, B. Hernando, On the injection of a K¨othe function space into L1(μ), Comment. Math., Prace Mat. 35 (1995), 49-60.
• [4] A. V. Bukhvalov, On an analytic representation of operators with abstract norm (in Russian), Izv. Vyss. Uceb. Zaved. 11 (1975), 21-32.
• [5] A. V. Bukhvalov, On an analytic representation of linear operators by vector-valued measurable functions, Izv. Vyss. Uceb. Zaved. 7 (1977), 21-32 (in Russian).
• [6] A. V. Bukhvalov, G. Ya. Lozanovskii, On sets closed in measure in spaces of measurable functions, Trans. Moscow Math. Soc. 2 (1978), 127-148.
• [7] J. Cerda, H. Hudzik, M. Masty lo, Geometric properties of K¨othe-Bochner spaces, Math. Proc. Cambridge Philos. Soc. 120 (1996), 521-533.
• [8] S. Chen, B. L. Lin, On strongly extreme points in K¨othe-Bochner spaces, Rocky Mountain J. Math. 27 (3) (1997), 1055-1063.
• [9] S. Chen, R. P luciennik, A note on H-points in K¨othe-Bochner spaces, Acta Math. Hungar. 94 (1-2) (2002), 59-66.
• [10] K. Feledziak, M. Nowak, Locally solid topologies on vector-valued function spaces, Collet. Math. 48, 4-6 (1997), 487-511.
• [11] V. A. Geuiler, L. V. Chubarova, Spaces of measurable vector-valued functions that do not contain the space l1, Mat. Zametki 34, no. 3 (1983), 425-430.
• [12] I. Halperin, Uniform convexity in function spaces, Duke Math. J. 21 (1954), 195-204.
• [13] L. V. Kantorovich, A. V. Akilov, Functional Analysis (in Russian), Nauka, Moscow 1984 (3rd ed).
• [14] P. K. Lin, K¨othe-Bochner function spaces, Birkh¨auser, Boston-Basel-Berlin, 2004.
• [15] M. Nowak, Duality theory of vector-valued function spaces I, Comment. Math., Prace Mat. 37 (1997), 195-215.
• [16] M. Nowak, Duality theory of vector-valued function spaces II, Comment. Math., Prace Mat. 37 (1997), 217-230.
• [17] M. Nowak, Duality theory of vector-valued function spaces III, Comment. Math., Prace Mat. 37 (1998), 101-108.
• [18] M. Nowak, Lebesgue topologies on vector-valued functions spaces, Math. Japonica 52, no. 2 (2000), 171-182.
• [19] M. Nowak, On some topological properties of vector-valued function spaces, Rocky Mtn. J. Math., 2007 (to appear).
• [20] A. W. Wilansky, Modern methods in topological vector spaces, McGrow-Hill, Inc., 1978.
• [21] A. C. Zaanen, Riesz spaces II, North-Holland Publ. Comp., Amsterdam, New York, Oxford, 1983.