Article

Full entry | PDF   (0.2 MB)
Keywords:
vector-valued continuous functions; strict topologies; locally solid topologies; weak-star compactness; vector measures
Summary:
Let $X$ be a completely regular Hausdorff space, $E$ a real normed space, and let $C_b(X,E)$ be the space of all bounded continuous $E$-valued functions on $X$. We develop the general duality theory of the space $C_b(X,E)$ endowed with locally solid topologies; in particular with the strict topologies $\beta_z(X,E)$ for $z=\sigma, \tau, t$. As an application, we consider criteria for relative weak-star compactness in the spaces of vector measures $M_z(X,E')$ for $z=\sigma, \tau, t$. It is shown that if a subset $H$ of $M_z(X,E')$ is relatively $\sigma(M_z(X,E'), C_b(X,E))$-compact, then the set $\operatorname{conv} (S(H))$ is still relatively $\sigma(M_z(X,E'), C_b(X,E))$-compact ($S(H)=$ the solid hull of $H$ in $M_z(X,E')$). A Mackey-Arens type theorem for locally convex-solid topologies on $C_b(X,E)$ is obtained.
References:
[AB$_1$] Aliprantis C.D., Burkinshaw O.: Locally Solid Riesz Spaces. Academic Press, New York, San Francisco, London, 1978. MR 0493242 | Zbl 1043.46003
[AB$_2$] Aliprantis C.D., Burkinshaw O.: Positive Operators. Academic Press, New York, 1985. MR 0809372 | Zbl 1098.47001
[F$_{{\pmb 1}}$] Fontenot R.A.: Strict topologies for vector-valued functions. Canad. J. Math. 26 (1974), 841-853. MR 0348463 | Zbl 0259.46037
[F$_{{\pmb 2}}$] Fontenot R.A.: Strict topologies for vector-valued functions, Corrigendum. Canad. J. Math. 27 5 (1975), 1183-1184. MR 0388057
[GJ] Gillman L., Jerison M.: Rings of Continuous Functions. Van Nostrand, Princeton, NJ, 1960. MR 0116199 | Zbl 0327.46040
[Kh] Khurana S.S.: Topologies on spaces of vector-valued continuous functions. Trans. Amer. Math. Soc. 241 (1978), 195-211. MR 0492297 | Zbl 0362.46035
[KhC] Khurana S.S., Chao S.A.: Strict topologies and $P$-spaces. Proc. Amer. Math. Soc. 61 (1976), 280-284. MR 0425603
[KhO$_{1}$] Khurana S.S., Othman S.I.: Convex compactness property in certain spaces of measures. Math. Ann. 279 (1987), 345-348. MR 0919510 | Zbl 0613.46041
[KhO$_{2}$] Khurana S.S., Othman S.I.: Grothendieck measures. J. London Math. Soc. (2) 39 (1989), 481-486. MR 1002460 | Zbl 0681.46030
[KhO$_{3}$] Khurana S.S., Othman S.I.: Completeness and sequential completeness in certain spaces of measures. Math. Slovaca 45 2 (1995), 163-170. MR 1357072 | Zbl 0832.46016
[KhV$_{1}$] Khurana S.S., Vielma J.: Strict topologies and perfect measures. Czechoslovak Math. J. 40 1 (1990), 1-7. MR 1032358
[KhV$_{2}$] Khurana S.S., Vielma J.: Weak sequential convergence and weak compactness in spaces of vector-valued continuous functions. J. Math. Anal. Appl. 195 (1995), 251-260. MR 1352821 | Zbl 0854.46032
[NR] Nowak M., Rzepka A.: Locally solid topologies on spaces of vector-valued continuous functions. Comment. Math. Univ. Carolinae 43 3 (2002), 473-483. MR 1920522 | Zbl 1068.46023
[S] Sentilles F.D.: Bounded continuous functions on a completely regular space. Trans. Amer. Math. Soc. 168 (1972), 311-336. MR 0295065 | Zbl 0244.46027
[V] Varadarajan V.S.: Measures on topological spaces. Amer. Math. Soc. Translations (2) 48 (1965), 161-220.
[Wh] Wheeler R.F.: Survey of Baire measures and strict topologies. Exposition Math. 2 (1983), 97-190. MR 0710569 | Zbl 0522.28009
[Wi] Wilansky A.: Modern Methods in Topological Vector Spaces. McGraw-Hill, New York, 1978. MR 0518316 | Zbl 0395.46001

Partner of