Geodesic
Strong topologies on vector-valued function spaces
Czechoslovak Mathematical Journal, Tome 50 (2000) no. 2, pp. 401-414 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $(X,\Vert \cdot \Vert _X)$ be a real Banach space and let $E$ be an ideal of $L^0$ over a $\sigma $-finite measure space $(Ø,\Sigma ,\mu )$. Let $(X)$ be the space of all strongly $\Sigma $-measurable functions $f\: Ø\rightarrow X$ such that the scalar function ${\widetilde{f}}$, defined by ${\widetilde{f}}(ø)=\Vert f(ø)\Vert _X$ for $ø\in Ø$, belongs to $E$. The paper deals with strong topologies on $E(X)$. In particular, the strong topology $\beta (E(X), E(X)^\sim _n)$ ($E(X)^\sim _n=$ the order continuous dual of $E(X)$) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.
Let $(X,\Vert \cdot \Vert _X)$ be a real Banach space and let $E$ be an ideal of $L^0$ over a $\sigma $-finite measure space $(Ø,\Sigma ,\mu )$. Let $(X)$ be the space of all strongly $\Sigma $-measurable functions $f\: Ø\rightarrow X$ such that the scalar function ${\widetilde{f}}$, defined by ${\widetilde{f}}(ø)=\Vert f(ø)\Vert _X$ for $ø\in Ø$, belongs to $E$. The paper deals with strong topologies on $E(X)$. In particular, the strong topology $\beta (E(X), E(X)^\sim _n)$ ($E(X)^\sim _n=$ the order continuous dual of $E(X)$) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.
Classification : 46A40, 46E30, 46E40
Keywords: vector valued function spaces; locally solid topologies; strong topologies; Mackey topologies; absolute weak topologies 
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Nowak, Marian. Strong topologies on vector-valued function spaces. Czechoslovak Mathematical Journal, Tome 50 (2000) no. 2, pp. 401-414. http://geodesic.mathdoc.fr/item/CMJ_2000_50_2_a14/

[AB$_1$] C.D. Aliprantis and O. Burkinshaw: Locally Solid Riesz Spaces. Academic Press, New York, San Francisco, London, 1978. | MR

[AB$_2$] C.D. Aliprantis and O. Burkinshaw: Positive Operators. Academic Press, Inc., 1985. | MR

[B$_1$] A.V. Bukhvalov: Vector-valued function spaces and tensor products. Siberian Math. J. 13 (1972), no. 6, 1229–1238. (Russian) | MR

[B$_2$] A.V. Bukhvalov: On an analytic representation of operators with abstract norm. Soviet. Math. Dokl. 14 (1973), 197–201. | Zbl

[B$_3$] A.V. Bukhvalov: On an analytic representation of operators with abstract norm. Izv. Vyssh. Ucebn. Zaved. Mat. 11 (1975), 21–32. (Russian) | MR

[B$_4$] A.V. Bukhvalov: On an analytic representation of linear operators by vector-valued measurable functions. Izv. Vyssh. Ucebn. Zaved. Mat. 7 (1977), 21–31. (Russian)

[CHM] J. Cerda, H. Hudzik, M. Mastyło: Geometric properties of Köthe-Bochner spaces. Math. Proc. Cambridge Philos. Soc. 120 (1996), 521–533. | DOI | MR

[DU] J. Diestel, J.J. Uhl Jr.: Vector Measures. Amer. Math. Soc., Math. Surveys 15, Providence, 1977. | MR

[FN] K. Feledziak, M. Nowak: Locally solid topologies on vector-valued function spaces. Collect. Math. 48, 4–6 (1997), 487–511. | MR

[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. | DOI | MR

[F] D.H. Fremlin: Topological Riesz Spaces and Measure Theory. Camb. Univ. Press, 1974. | MR | Zbl

[G] D.A. Gregory: Some basic properties of vector sequence spaces. J. Reine Angew. Math. 237 (1969), 26–38. | MR

[KA] L.V. Kantorovitch, G.P. Akilov: Functional Analysis. 3$^{rd}$ ed., Nauka, Moscow, 1984. (Russian) | MR

[K] G. Köthe: Topological Vector Spaces I. Springer-Verlag, Berlin, Heidelberg, New York, 1983. | MR

[M] A.L. Macdonald: Vector valued Köthe function spaces I. Illinois J. Math. 17 (1973), 533–545. | DOI | MR | Zbl

[MR] L.C. Moore, J.C. Reber: Mackey topologies which are locally convex Riesz topologies. Duke Math. J. 39 (1972), 105–119. | DOI | MR

[N$_1$] M. Nowak: Duality theory of vector valued function spaces I. Comment. Math. 37 (1997), 195–215. | MR | Zbl

[N$_2$] M. Nowak: Duality theory of vector–valued function spaces III. Comment. Math. 38 (1998), 101–108. | MR | Zbl

[PC] N. Phuong-Các: Generalized Köthe function spaces I. Math. Proc. Cambridge Philos. Soc. 65 (1969), 601–611. | DOI | MR

[Ro] A.P. Robertson, W.J. Robertson: Topological Vector Spaces. Cambridge, 1973. | MR

[R] R.C. Rosier: Dual spaces of certain vector sequence spaces. Pacific J. Math. 46 (1973), 487–501. | DOI | MR | Zbl

[W] J.H. Webb: Sequential convergence in locally convex spaces. Math. Proc. Cambridge Philos. Soc. 64 (1968), 341–364. | DOI | MR | Zbl

[We] R. Welland: On Köthe spaces. Trans. Amer. Math. Soc. 112 (1964), 267–277. | DOI | MR | Zbl

[Wi] A. Wilansky: Modern Methods in Topological Vector Spaces. Mc Graw-Hill, Inc., 1978. | MR | Zbl

[V] B.Z. Vulikh: Introduction to the Theory of Partially Ordered Spaces. Wolter-Hoordhoff, Groningen, Netherlands, 1967. | MR | Zbl