Uniformly μ -continuous topologies on Köthe-Bochner spaces and Orlicz-Bochner spaces

Krzysztof Feledziak

Commentationes Mathematicae Universitatis Carolinae (1998)

  • Volume: 39, Issue: 3, page 453-468
  • ISSN: 0010-2628

Abstract

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Some class of locally solid topologies (called uniformly μ -continuous) on Köthe-Bochner spaces that are continuous with respect to some natural two-norm convergence are introduced and studied. A characterization of uniformly μ -continuous topologies in terms of some family of pseudonorms is given. The finest uniformly μ -continuous topology 𝒯 I ϕ ( X ) on the Orlicz-Bochner space L ϕ ( X ) is a generalized mixed topology in the sense of P. Turpin (see [11, Chapter I]).

How to cite

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Feledziak, Krzysztof. "Uniformly $\mu $-continuous topologies on Köthe-Bochner spaces and Orlicz-Bochner spaces." Commentationes Mathematicae Universitatis Carolinae 39.3 (1998): 453-468. <http://eudml.org/doc/248263>.

@article{Feledziak1998,
abstract = {Some class of locally solid topologies (called uniformly $\mu $-continuous) on Köthe-Bochner spaces that are continuous with respect to some natural two-norm convergence are introduced and studied. A characterization of uniformly $\mu $-continuous topologies in terms of some family of pseudonorms is given. The finest uniformly $\mu $-continuous topology $\mathcal \{T\}^\varphi _I(X)$ on the Orlicz-Bochner space $L^\varphi (X)$ is a generalized mixed topology in the sense of P. Turpin (see [11, Chapter I]).},
author = {Feledziak, Krzysztof},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Orlicz spaces; Orlicz-Bochner spaces; Köthe-Bochner spaces; locally solid topologies; generalized mixed topologies; uniformly $\mu $-continuous topologies; inductive limit topologies; Köthe-Bochner spaces; Orlicz-Bochner spaces; locally solid topology; generalized mixed topology; inductive limit topology; uniformly -continuous topology},
language = {eng},
number = {3},
pages = {453-468},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Uniformly $\mu $-continuous topologies on Köthe-Bochner spaces and Orlicz-Bochner spaces},
url = {http://eudml.org/doc/248263},
volume = {39},
year = {1998},
}

TY - JOUR
AU - Feledziak, Krzysztof
TI - Uniformly $\mu $-continuous topologies on Köthe-Bochner spaces and Orlicz-Bochner spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1998
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 39
IS - 3
SP - 453
EP - 468
AB - Some class of locally solid topologies (called uniformly $\mu $-continuous) on Köthe-Bochner spaces that are continuous with respect to some natural two-norm convergence are introduced and studied. A characterization of uniformly $\mu $-continuous topologies in terms of some family of pseudonorms is given. The finest uniformly $\mu $-continuous topology $\mathcal {T}^\varphi _I(X)$ on the Orlicz-Bochner space $L^\varphi (X)$ is a generalized mixed topology in the sense of P. Turpin (see [11, Chapter I]).
LA - eng
KW - Orlicz spaces; Orlicz-Bochner spaces; Köthe-Bochner spaces; locally solid topologies; generalized mixed topologies; uniformly $\mu $-continuous topologies; inductive limit topologies; Köthe-Bochner spaces; Orlicz-Bochner spaces; locally solid topology; generalized mixed topology; inductive limit topology; uniformly -continuous topology
UR - http://eudml.org/doc/248263
ER -

References

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  1. Aliprantis C.D., Burkinshaw O., Locally Solid Riesz Spaces, Academic Press, New York, 1978. Zbl1043.46003MR0493242
  2. Bukhvalov A.V., On an analytic representation of operators with abstract norm (in Russian), Izv. Vyssh. Uchebn. Zaved. 11 (1975), 21-32. (1975) MR0470746
  3. Feledziak K., Nowak M., Locally solid topologies on vector valued function spaces, Collect. Math., to appear. Zbl0912.46028MR1602576
  4. Köthe G., Topological Vector Spaces I, Springer-Verlag, Berlin, Heidelberg, New York, 1983. MR0248498
  5. Krasnoselskii M., Rutickii Ya.B., Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961. MR0126722
  6. Luxemburg W.A., Banach Function Spaces, Delft, 1955. Zbl0162.44701MR0072440
  7. Nowak M., On two linear topologies on Orlicz spaces L * ϕ , I, Comment. Math. 23 (1983), 71-84. (1983) Zbl0602.46020MR0709174
  8. Nowak M., Inductive limit of a sequence of balanced topological spaces in Orlicz spaces L E * ϕ ( μ ) , Comment. Math. 25 (1985), 295-313. (1985) Zbl0607.46021MR0844647
  9. Nowak M., On some linear topology in Orlicz spaces L E * ϕ ( μ ) , I, Comment. Math. 26 (1986), 51-68. (1986) MR0857526
  10. Nowak M., A generalized mixed topology on Orlicz spaces, Revista Matematica 7 (1) (1994), 27-56. (1994) Zbl0806.46032MR1277329
  11. Turpin P., Convexités dans les espaces vectoriels topologiques généraux, Dissertationes Math. 131 (1976), 221 pp. (1976) Zbl0331.46001MR0423044
  12. Wiweger A., Linear spaces with mixed topology, Studia Math. 20 (1961), 47-68. (1961) Zbl0097.31301MR0133664

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