Strong topologies on vector-valued function spaces
Tóm tắt
Let
$$(X,\left\| \cdot \right\|_X )$$
be a real Banach space and let E be an ideal of L
0 over a σ-finite measure space (Ω, Σ, μ). Let (X) be the space of all strongly Σ-measurable functions f: Ω → X such that the scalar function
$${\tilde f}$$
, defined by
$$\tilde f(\omega ) = \left\| {f(\omega )} \right\|_X {\text{ for }}\omega \in \Omega $$
, belongs to E. The paper deals with strong topologies on E(X). In particular, the strong topology
$$\beta ({\rm E}(X),{\rm E}(X)\user1{\tilde n)(}{\rm E}(X)\user1{\tilde n} = $$
the order continuous dual of E(X)) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.
Tài liệu tham khảo
[AB1]C.D. Aliprantis and O. Burkinshaw: Locally Solid Riesz Spaces. Academic Press, New York, San Francisco, London, 1978.
[AB2]C.D. Aliprantis and O. Burkinshaw: Positive Operators. Academic Press, Inc., 1985.
[B1]A. V. Bukhvalov: Vector-valued function spaces and tensor products. Siberian Math. J. 13 (1972), no. 6, 1229–1238. (In Russian.)
[B2]A. V. Bukhvalov: On an analytic representation of operators with abstract norm. Soviet. Math. Dokl. 14 (1973), 197–201.
[B3]A. V. Bukhvalov: On an analytic representation of operators with abstract norm. Izv. Vyssh. Ucebn. Zaved. Mat. 11 (1975), 21–32. (In Russian.)
[B4]A. V. Bukhvalov: On an analytic representation of linear operators by vector-valued measurable functions. Izv. Vyssh. Ucebn. Zaved. Mat. 7 (1977), 21–31. (In Russian.)
[CHM]J. Cerda, H. Hudzik, M. Mastyło: Geometric properties of Köthe-Bochner spaces. Math. Proc. Cambridge Philos. Soc. 120 (1996), 521–533.
[DU]J. Diestel, J.J. Uhl Jr.: Vector Measures. Amer. Math. Soc., Math. Surveys 15, Providence, 1977.
[FN]K. Feledziak, M. Nowak: Locally solid topologies on vector-valued function spaces. Collect. Math. 48,4–6 (1997), 487–511.
[FPS]M. Florencio, P.J. Paúl annd C. Sáez: Duals of vector-valued Köthe function spaces. Math. Proc. Cambridge Philos. Soc. 112 (1992), 165–174.
[F]D.H. Fremlin: Topological Riesz Spaces and Measure Theory. Camb. Univ. Press, 1974.
[G]D.A. Gregory: Some basic properties of vector sequence spaces. J. Reine Angew. Math. 237 (1969), 26–38.
[KA]L.V. Kantorovitch, G.P. Akilov: Functional Analysis. 3rd ed., Nauka, Moscow, 1984. (In Russian.)
[K]G. Köthe: Topological Vector Spaces I. Springer-Verlag, Berlin, Heidelberg, New York, 1983.
[M]A.L. Macdonald: Vector valued Köthe function spaces I. Illinois J. Math. 17 (1973), 533–545; II. Illinois J. Math. 17 (1973), 546–557; III. Illinois J. Math. 18 (1974), 136–146.
[MR]L.C. Moore, J.C. Reber: Mackey topologies which are locally convex Riesz topologies. Duke Math. J. 39 (1972), 105–119.
[N1]M. Nowak: Duality theory of vector valued function spaces I. Comment. Math. 37 (1997), 195–215.
[N2]M. Nowak: Duality theory of vector-valued function spaces III. Comment. Math. 38 (1998), 101–108.
[PC]N. Phuong-Các: Generalized Köthe function spaces I. Math. Proc. Cambridge Philos. Soc. 65 (1969), 601–611.
[Ro]A.P. Robertson, W.J. Robertson: Topological Vector Spaces. Cambridge, 1973.
[R]R.C. Rosier: Dual spaces of certain vector sequence spaces. Pacific J. Math. 46 (1973), 487–501.
[W]J.H. Webb: Sequential convergence in locally convex spaces. Math. Proc. Cambridge Philos. Soc. 64 (1968), 341–364.
[We]R. Welland: On Köthe spaces. Trans. Amer. Math. Soc. 112 (1964), 267–277.
[Wi]A. Wilansky: Modern Methods in Topological Vector Spaces. Mc Graw-Hill, Inc., 1978.
[V]B.Z. Vulikh: Introduction to the Theory of Partially Ordered Spaces. Wolter-Hoord-hoff, Groningen, Netherlands, 1967.