Topological properties of some spaces of continuous operators

Marian Nowak

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2016)

  • Volume: 36, Issue: 1, page 79-86
  • ISSN: 1509-9407

Abstract

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Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let C b ( X , E ) be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study topological properties of the space L β ( C b ( X , E ) , F ) of all ( β , | | · | | F ) -continuous linear operators from C b ( X , E ) to F, equipped with the topology τ s of simple convergence. If X is a locally compact paracompact space (resp. a P-space), we characterize τ s -compact subsets of L β ( C b ( X , E ) , F ) in terms of properties of the corresponding sets of the representing operator-valued Borel measures. It is shown that the space ( L β ( C b ( X , E ) , F ) , τ s ) is sequentially complete if X is a locally compact paracompact space.

How to cite

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Marian Nowak. "Topological properties of some spaces of continuous operators." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 36.1 (2016): 79-86. <http://eudml.org/doc/286913>.

@article{MarianNowak2016,
abstract = {Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let $C_b(X,E)$ be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study topological properties of the space $L_\{β\}(C_\{b\}(X,E),F)$ of all $(β,||·||_\{F\})$-continuous linear operators from $C_\{b\}(X,E)$ to F, equipped with the topology $τ_\{s\}$ of simple convergence. If X is a locally compact paracompact space (resp. a P-space), we characterize $τ_\{s\}$-compact subsets of $L_\{β\}(C_\{b\}(X,E),F)$ in terms of properties of the corresponding sets of the representing operator-valued Borel measures. It is shown that the space $(L_\{β\}(C_\{b\}(X,E),F),τ_\{s\})$ is sequentially complete if X is a locally compact paracompact space.},
author = {Marian Nowak},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {spaces of vector-valued continuous functions; strict topologies; operator measures; topology of simple convergence; continuous operators},
language = {eng},
number = {1},
pages = {79-86},
title = {Topological properties of some spaces of continuous operators},
url = {http://eudml.org/doc/286913},
volume = {36},
year = {2016},
}

TY - JOUR
AU - Marian Nowak
TI - Topological properties of some spaces of continuous operators
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2016
VL - 36
IS - 1
SP - 79
EP - 86
AB - Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let $C_b(X,E)$ be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study topological properties of the space $L_{β}(C_{b}(X,E),F)$ of all $(β,||·||_{F})$-continuous linear operators from $C_{b}(X,E)$ to F, equipped with the topology $τ_{s}$ of simple convergence. If X is a locally compact paracompact space (resp. a P-space), we characterize $τ_{s}$-compact subsets of $L_{β}(C_{b}(X,E),F)$ in terms of properties of the corresponding sets of the representing operator-valued Borel measures. It is shown that the space $(L_{β}(C_{b}(X,E),F),τ_{s})$ is sequentially complete if X is a locally compact paracompact space.
LA - eng
KW - spaces of vector-valued continuous functions; strict topologies; operator measures; topology of simple convergence; continuous operators
UR - http://eudml.org/doc/286913
ER -

References

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  1. [1] N. Bourbaki, Elements of Mathematics, Topological Vector Spaces, Chap. 1-5 (Springer, Berlin, 1987). doi: 10.1007/978-3-642-61715-7 Zbl0622.46001
  2. [2] R.C. Buck, Bounded contiuous functions on a locally compact space, Michigan Math. J. 5 (1958), 95-104. doi: 10.1307/mmj/1028998054 Zbl0087.31502
  3. [3] S. Choo, Strict topology on spaces of continuous vector-valued functions, Can. J. Math. 31 (4) (1979), 890-896. doi: 10.4153/CJM-1979-084-9 Zbl0393.46031
  4. [4] J.B. Cooper, The strict topology and spaces with mixed topologies, Proc. Amer. Math. Soc. 30 (3) (1971), 583-592. doi: 10.1090/S0002-9939-1971-0284789-2 Zbl0225.46004
  5. [5] N. Dinculeanu, Vector Measures (Pergamon Press, New York, 1967). doi: 10.1016/b978-1-4831-9762-3.50004-4 
  6. [6] D. Fontenot, Strict topologies for vector-valued functions, Canad. J. Math. 26 (4) (1974), 841-853. doi: 10.4153/CJM-1974-079-1 Zbl0259.46037
  7. [7] R.K. Goodrich, A Riesz representation theorem, Proc. Amer. Math. Soc. 24 (1970), 629-636. doi: 10.1090/S0002-9939-1970-0415386-2 Zbl0193.42002
  8. [8] L. Gillman and M. Henriksen, Concerning rings of continuous functions, Trans. Amer. Math. Soc. 77 (1954), 340-362. doi: 10.1090/S0002-9947-1954-0063646-5 Zbl0058.10003
  9. [9] L.A. Khan, The strict topology on a space of vector-valued functions, Proc. Edinburgh Math. Soc. 22 (1) (1979), 35-41. doi: 10.1017/S0013091500027784 Zbl0429.46023
  10. [10] L.A. Khan and K. Rowlands, On the representation of strictly continuous linear functionals, Proc. Edinburgh Math. Soc. 24 (1981), 123-130. doi: 10.1017/S0013091500006428 Zbl0459.46026
  11. [11] S.S. Khurana, Topologies on spaces of vector-valued continuous functions, Trans. Amer. Math. Soc. 241 (1978), 195-211. doi: 10.1090/S0002-9947-1978-0492297-X Zbl0335.46017
  12. [12] S.S. Khurana and S.A. Choo, Strict topology and P-spaces, Proc. Amer. Math. Soc. 61 (1976), 280-284. doi: 10.2307/2041326 Zbl0322.46052
  13. [13] S.S. Khurana and S.I. Othman, Completeness and sequential completeness in certain spaces of measures, Math. Slovaca 45 (2) (1995), 163-170. Zbl0832.46016
  14. [14] M. Nowak, A Riesz representation theory for completely regular Hausdorff spaces and its applications, Open Math., (in press). 
  15. [15] W. Ruess, [Weakly] compact operators and DF-spaces, Pacific J. Math. 98 (1982), 419-441. doi: 10.2140/pjm.1982.98.419 
  16. [16] H. Schaeffer and X.-D. Zhang, On the Vitali-Hahn-Saks theorem, Operator Theory, Adv. Appl. 75 (Birkhäuser, Basel, 1995), 289-297. 
  17. [17] J. Schmets and J. Zafarani, Strict topologies and (gDF)-spaces, Arch. Math. 49 (1987), 227-231. doi: 10.1007/BF01271662 Zbl0615.46032
  18. [18] R. Wheeler, The strict topology for P-spaces, Proc. Amer. Math. Soc. 41 (2) (1973), 466-472. doi: 10.2307/2039115 Zbl0272.46018
  19. [19] A. Wiweger, Linear spaces with mixed topology, Studia Math. 20 (1961), 47-68. Zbl0097.31301

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