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1 Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

1 2004

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Generalized mixed topology on F-normed function spaces

Kasior E., Nowak M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2004

Vol. 44, nr 1

55--66

Let (X, ||•||) be a F-normed function space over a σ-finite measure space (Ω, Σ, μ) and let ||•||0 denote the usual F-norm on L0 that generates the convergence in measure on subsets of finite measures. In X a natural two-normed convergence can be defined as follows: a sequence (xn) in X is said to be γ-convergent to x ϵ X whenever || xn - x||0 → 0 and supn||xn|| < ∞. In this paper we study locally solid topologies on X satisfying the continuity property with respect to this γ-convergence in X. We call such topologies "uniformly Lebesgue". These investigations are closely related to the theory of generalized inductive limit topologies in the sense of Turpin. In particular we show that a generalized mixed topology γT(Tφ, T0|Lφ) on the Orlicz space Lφ (φ is not assumed to be convex) is the finest uniformly Lebesgue topology on Lφ. Moreover, we characterize γφ-linear functionals on Lφ.