Locally solid topologies on spaces of vector-valued continuous functions
Nowak, Marian ; Rzepka, Aleksandra
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002), p. 473-483 / Harvested from Czech Digital Mathematics Library
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Let $X$ be a completely regular Hausdorff space and $E$ a real normed space. We examine the general properties of locally solid topologies on the space $C_b(X,E)$ of all $E$-valued continuous and bounded functions from $X$ into $E$. The mutual relationship between locally solid topologies on $C_b(X,E)$ and $C_b(X)$ $(=C_b(X,\Bbb R))$ is considered. In particular, the mutual relationship between strict topologies on $C_b(X)$ and $C_b(X,E)$ is established. It is shown that the strict topology $\beta _\sigma(X,E)$ (respectively $\beta _\tau(X,E)$) is the finest $\sigma $-Dini topology (respectively Dini topology) on $C_b(X,E)$. A characterization of $\sigma $-Dini and Dini topologies on $C_b(X,E)$ in terms of their topological duals is given.

Publié le : 2002-01-01
Classification:  46A03,  46E05,  46E10,  46E40,  47A70 
@article{119336,
     author = {Marian Nowak and Aleksandra Rzepka},
     title = {Locally solid topologies on spaces of vector-valued continuous functions},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {43},
     year = {2002},
     pages = {473-483},
     zbl = {1068.46023},
     mrnumber = {1920522},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119336}
}

Nowak, Marian; Rzepka, Aleksandra. Locally solid topologies on spaces of vector-valued continuous functions. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 473-483. http://gdmltest.u-ga.fr/item/119336/

Aguayo-Garrido J. Strict topologies on spaces of continuous functions and $u$-additive measure spaces, J. Math. Anal. Appl. 220 (1998), 77-89. (1998) | MR 1612071 | Zbl 0914.46031

Aliprantis C.D.; Burkinshaw O. Locally Solid Topologies, Academic Press, New York, San Francisco, London, 1978. | MR 0493242 | Zbl 0436.46009

Fontenot R.A. Strict topologies for vector-valued functions, Canad. J. Math. 26 (1974), 841-853. (1974) | MR 0348463 | Zbl 0259.46037

Katsaras A.K. Spaces of vector measures, Trans. Amer. Math. Soc. 206 (1975), 313-328. (1975) | MR 0365111 | Zbl 0275.46029

Katsaras A.K. Some locally convex spaces of continuous vector-valued functions over a completely regular space and their duals, Trans. Amer. Math. Soc. 216 (1976), 367-387. (1976) | MR 0390733 | Zbl 0317.46031

Katsaras A.K. Locally convex topologies on spaces of continuous vector functions, Math. Nachr. 71 (1976), 211-226. (1976) | MR 0458143 | Zbl 0281.46032

Khurana S.S. Topologies on spaces of vector-valued continuous functions, Trans. Amer. Math. Soc. 241 (1978), 195-211. (1978) | MR 0492297 | Zbl 0362.46035

Khurana S.S.; Othman S.I. Grothendieck measures, J. London Math. Soc. (2) 39 (1989), 481-486. (1989) | MR 1002460 | Zbl 0681.46030

Khurana S.S.; Vielma J.E. Strict topology and perfect measures, Czechoslovak Math. J. 40 (1990), 1-7. (1990) | MR 1032358 | Zbl 0711.28002

Khurana S.S.; Vielma J.E. Weak sequential convergence and weak compactness in spaces of vector-valued continuous functions, J. Math. Anal. Appl. 195 (1995), 251-260. (1995) | MR 1352821 | Zbl 0854.46032

Sentilles F.D. Bounded continous functions on a completely regular space, Trans. Amer. Math. Soc. 168 (1972), 311-336. (1972) | MR 0295065

Wheeler R.F. Survey of Baire measures and strict topologies, Exposition Math. 2 (1983), 97-190. (1983) | MR 0710569 | Zbl 0522.28009

Wheeler R.F. The strict topology, separable measures, and paracompactness, Pacific J. Math. 47 (1973), 287-302. (1973) | MR 0341047 | Zbl 0244.46028