# Article

Full entry | PDF   (0.2 MB)
Keywords:
Orlicz spaces; Orlicz-Bochner spaces; Köthe-Bochner spaces; locally solid topologies; generalized mixed topologies; uniformly $\mu$-continuous topologies; inductive limit topologies
Summary:
Some class of locally solid topologies (called uniformly $\mu$-continuous) on \linebreak 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 $\Cal T^\varphi_I(X)$ on the Orlicz-Bochner space $L^\varphi(X)$ is a generalized mixed topology in the sense of P. Turpin (see [11, Chapter I]).
References:
[1] Aliprantis C.D., Burkinshaw O.: Locally Solid Riesz Spaces. Academic Press, New York, 1978. MR 0493242 | Zbl 1043.46003
[2] Bukhvalov A.V.: On an analytic representation of operators with abstract norm (in Russian). Izv. Vyssh. Uchebn. Zaved. 11 (1975), 21-32. MR 0470746
[3] Feledziak K., Nowak M.: Locally solid topologies on vector valued function spaces. Collect. Math., to appear. MR 1602576 | Zbl 0912.46028
[4] Köthe G.: Topological Vector Spaces I. Springer-Verlag, Berlin, Heidelberg, New York, 1983. MR 0248498
[5] Krasnoselskii M., Rutickii Ya.B.: Convex Functions and Orlicz Spaces. P. Noordhoff Ltd., Groningen, 1961. MR 0126722
[6] Luxemburg W.A.: Banach Function Spaces. Delft, 1955. MR 0072440 | Zbl 0162.44701
[7] Nowak M.: On two linear topologies on Orlicz spaces $L^{*\varphi}$, I. Comment. Math. 23 (1983), 71-84. MR 0709174 | Zbl 0602.46020
[8] Nowak M.: Inductive limit of a sequence of balanced topological spaces in Orlicz spaces $L^{*\varphi}_E(\mu)$. Comment. Math. 25 (1985), 295-313. MR 0844647 | Zbl 0607.46021
[9] Nowak M.: On some linear topology in Orlicz spaces $L^{*\varphi}_E(\mu)$, I. Comment. Math. 26 (1986), 51-68. MR 0857526
[10] Nowak M.: A generalized mixed topology on Orlicz spaces. Revista Matematica 7 (1) (1994), 27-56. MR 1277329 | Zbl 0806.46032
[11] Turpin P.: Convexités dans les espaces vectoriels topologiques généraux. Dissertationes Math. 131 (1976), 221 pp. MR 0423044 | Zbl 0331.46001
[12] Wiweger A.: Linear spaces with mixed topology. Studia Math. 20 (1961), 47-68. MR 0133664 | Zbl 0097.31301

Partner of