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Keywords:
vector-valued continuous functions; strict topologies; locally solid topologies; Dini-topologies; strong Mackey space; $\sigma$-additive operators; $\tau$-additive operators
Summary:
Let $X$ be a completely regular Hausdorff space, $E$ a real Banach space, and let $C_b(X,E)$ be the space of all $E$-valued bounded continuous functions on $X$. We study linear operators from $C_b(X,E)$ endowed with the strict topologies $\beta_z$ $(z=\sigma,\tau,\infty,g)$ to a real Banach space $(Y,\|\cdot\|_Y)$. In particular, we derive Banach-Steinhaus type theorems for $(\beta_z,\|\cdot\|_Y)$ continuous linear operators from $C_b(X,E)$ to $Y$. Moreover, we study $\sigma$-additive and $\tau$-additive operators from $C_b(X,E)$ to $Y$.
References:
[AB] Aliprantis C.D., Burkinshaw O.: Positive Operators. Academic Press, New York, 1985. MR 0809372 | Zbl 1098.47001
[F] Fontenot R.A.: Strict topologies for vector-valued functions. Canad. J. Math. 26 (1974), 841--853. DOI 10.4153/CJM-1974-079-1 | MR 0348463 | Zbl 0259.46037
[K] Khurana S.S.: Topologies on spaces of vector-valued continuous functions. Trans. Amer. Math. Soc. 241 (1978), 195--211. DOI 10.1090/S0002-9947-1978-0492297-X | MR 0492297 | Zbl 0487.46014
[KO$_1$] Khurana S.S., Othman S.I.: Convex compactnes property in some spaces of measures. Math. Ann. 279 (1987), 345--348. DOI 10.1007/BF01461727 | MR 0919510
[KO$_2$] Khurana S.S., Othman S.I.: Grothendieck measures. J. London Math. Soc. (2) 39 (1989), 481--486. DOI 10.1112/jlms/s2-39.3.481 | MR 1002460 | Zbl 0681.46030
[KO$_3$] Khurana S.S., Othman S.I.: Completeness and sequential completeness in certain spaces of measures. Math. Slovaca 45 (1995), no. 2, 163--170. MR 1357072 | Zbl 0832.46016
[KV] 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. DOI 10.1006/jmaa.1995.1353 | 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 (2002), no. 3, 473--483. MR 1920522 | Zbl 1068.46023
[SZ] Schaefer H., Zhang X.-D.: On the Vitali-Hahn-Saks theorem. Oper. Theory Adv. Appl., 75, Birkhäuser, Basel, 1995, pp. 289--297. MR 1322508 | Zbl 0830.28007

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