Previous |  Up |  Next


vector valued function spaces; locally solid topologies; strong topologies; Mackey topologies; absolute weak 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.
[AB$_1$] C.D. Aliprantis and O. Burkinshaw: Locally Solid Riesz Spaces. Academic Press, New York, San Francisco, London, 1978. MR 0493242
[AB$_2$] C.D. Aliprantis and O. Burkinshaw: Positive Operators. Academic Press, Inc., 1985. MR 0809372
[B$_1$] A.V. Bukhvalov: Vector-valued function spaces and tensor products. Siberian Math. J. 13 (1972), no. 6, 1229–1238. (Russian) MR 0358342
[B$_2$] A.V. Bukhvalov: On an analytic representation of operators with abstract norm. Soviet. Math. Dokl. 14 (1973), 197–201. Zbl 0283.47028
[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 0470746
[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 10.1017/S0305004100075058 | MR 1388204
[DU] J. Diestel, J.J. Uhl Jr.: Vector Measures. Amer. Math. Soc., Math. Surveys 15, Providence, 1977. MR 0453964
[FN] K. Feledziak, M. Nowak: Locally solid topologies on vector-valued function spaces. Collect. Math. 48, 4–6 (1997), 487–511. MR 1602576
[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 10.1017/S0305004100070845 | MR 1162941
[F] D.H. Fremlin: Topological Riesz Spaces and Measure Theory. Camb. Univ. Press, 1974. MR 0454575 | Zbl 0273.46035
[G] D.A. Gregory: Some basic properties of vector sequence spaces. J. Reine Angew. Math. 237 (1969), 26–38. MR 0251497
[KA] L.V. Kantorovitch, G.P. Akilov: Functional Analysis. 3$^{rd}$ ed., Nauka, Moscow, 1984. (Russian) MR 0788496
[K] G. Köthe: Topological Vector Spaces I. Springer-Verlag, Berlin, Heidelberg, New York, 1983. MR 0248498
[M] A.L. Macdonald: Vector valued Köthe function spaces I. Illinois J. Math. 17 (1973), 533–545. MR 0333662 | Zbl 0271.46034
[MR] L.C. Moore, J.C. Reber: Mackey topologies which are locally convex Riesz topologies. Duke Math. J. 39 (1972), 105–119. DOI 10.1215/S0012-7094-72-03915-4 | MR 0295045
[N$_1$] M. Nowak: Duality theory of vector valued function spaces I. Comment. Math. 37 (1997), 195–215. MR 1608189 | Zbl 0908.46023
[N$_2$] M. Nowak: Duality theory of vector–valued function spaces III. Comment. Math. 38 (1998), 101–108. MR 1672244 | Zbl 0972.46025
[PC] N. Phuong-Các: Generalized Köthe function spaces I. Math. Proc. Cambridge Philos. Soc. 65 (1969), 601–611. DOI 10.1017/S030500410000339X | MR 0248499
[Ro] A.P. Robertson, W.J. Robertson: Topological Vector Spaces. Cambridge, 1973. MR 0350361
[R] R.C. Rosier: Dual spaces of certain vector sequence spaces. Pacific J. Math. 46 (1973), 487–501. DOI 10.2140/pjm.1973.46.487 | MR 0328544 | Zbl 0263.46009
[W] J.H. Webb: Sequential convergence in locally convex spaces. Math. Proc. Cambridge Philos. Soc. 64 (1968), 341–364. DOI 10.1017/S0305004100042900 | MR 0222602 | Zbl 0157.20202
[We] R. Welland: On Köthe spaces. Trans. Amer. Math. Soc. 112 (1964), 267–277. DOI 10.2307/1994294 | MR 0172110 | Zbl 0122.11501
[Wi] A. Wilansky: Modern Methods in Topological Vector Spaces. Mc Graw-Hill, Inc., 1978. MR 0518316 | Zbl 0395.46001
[V] B.Z. Vulikh: Introduction to the Theory of Partially Ordered Spaces. Wolter-Hoordhoff, Groningen, Netherlands, 1967. MR 0224522 | Zbl 0186.44601
Partner of
EuDML logo