Completeness and sequential completeness in certain spaces of measures. (English) Zbl 0832.46016

Khurana, Surjit Singh; Othman, Sadoon Ibrahim

Math. Slovaca 45, No. 2, 163-170 (1995).

Summary: Let \(X\) be a completely regular Hausdorff space, \(E\) a Banach space over \(K\), the field of real or complex numbers, \(C(X, E)\) (\(C(X)\) if \(E= K\)) the space of all \(E\)-valued continuous functions on \(X\), and \(C_b(X, E)\) (\(C_b(X)\) if \(E= K\)) the space of all \(E\)-valued bounded continuous functions on \(X\). Put \(F_z= (C_b(X, E), \beta_z)\) (\(\beta_z\) the so called strict topologies), and \(F= (C(X, E), \beta_{\infty C})\).
It is proved that \((F_z', \sigma(F_z', F_z))\) is sequentially complete for \(z= \sigma, \infty, g\); if, in addition \(X\) is meta-compact and normal, then the result is also true for \(z= \tau\). Also it is proved that \((F', \sigma(F', F))\) is sequentially complete. For the Mackey topology it is proved that \((F_z', \tau(F_z', F_z))\) is complete for \(z= \sigma, \infty, g\) and for \(z= \tau(t)\) it is complete if and only if \(M_g(X)= M_\tau(x)\) \((M_t(X))\). Further, it is proved that \((F', \tau(F', F))\) is complete. Some additional results are proved for sequential convergence.

MSC:

46E10	Topological linear spaces of continuous, differentiable or analytic functions
47A70	(Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
46E40	Spaces of vector- and operator-valued functions
46G10	Vector-valued measures and integration

Keywords:

completely regular Hausdorff space; strict topologies; sequentially complete; meta-compact; Mackey topology

References:

[1]	BERRUYER, IVOL B.: L’espace M(T). Comptes Rendus 275 (1972), 33-36. · Zbl 0244.46030
[2]	FONTENOT R. A.: Strict topologies for vector-valued functions. Canad. J. Math. 26 (1974), 841-853. · Zbl 0259.46037 · doi:10.4153/CJM-1974-079-1
[3]	KATSARAS A. K.: Spaces of vector measures. Trans. Amer. Math. Soc. 206 (1975), 313-328. · Zbl 0275.46029 · doi:10.2307/1997159
[4]	KATSARAS A. K.: Locally convex topologies on spaces of continuous vector functions. Math. Nachr. 71 (1976), 211-226. · Zbl 0281.46032 · doi:10.1002/mana.19760710117
[5]	KHURANA S. S.: Topologies on spaces of continuous vector-valued functions. Trans. Amer. Math. Soc. 241 (1978), 195-211. · Zbl 0335.46017 · doi:10.2307/1998840
[6]	KHURANA S. S.: Topologies on spaces of continuous vector-valued functions II. Math. Ann. 234 (1978), 159-166. · Zbl 0362.46035 · doi:10.1007/BF01420966
[7]	KHURANA S. S., OTHMAN S.: Convex compact property in certain spaces of measures. Math. Ann. 279 (1987), 345-348. · Zbl 0613.46041 · doi:10.1007/BF01461727
[8]	KHURANA S. S., OTHMAN S.: Grothendieck measures. J. London Math. Soc. (2) 39 (1989), 481-486. · Zbl 0681.46030 · doi:10.1112/jlms/s2-39.3.481
[9]	KIRK R. B.: Complete topologies on spaces of Baire measures. Trans. Amer. Math. Soc. 184 (1973), 1-29. · Zbl 0296.60005 · doi:10.2307/1996396
[10]	MORAN W.: Measures on metacompact spaces. Proc. London Math. Soc. (3) 20 (1970), 507-524. · Zbl 0199.37802 · doi:10.1112/plms/s3-20.3.507
[11]	SCHAEFFER H. H.: Topological Vector Spaces. Springer Verlag, New York, 1986.
[12]	SENTILLES F. D.: Bounded continuous functions on completely regular spaces. Trans. Amer. Math. Soc. 168 (1972), 311-336. · Zbl 0244.46027 · doi:10.2307/1996178
[13]	WHEELER R. F.: Survey of Baire measures and strict topologies. Exposition Math. 2 (1983), 97-190. · Zbl 0522.28009
[14]	VARADARAJAN V. S.: Measures on topological spaces. Amer. Math. Soc. Transl. Ser. 2, 48, 1965, pp. 161-220.