Hyperbolic topology of normed linear spaces
نویسندگان
چکیده
In a previous paper [6], the authors introduced the hyperbolic topology on a metric space, which is weaker than the metric topology and naturally derived from the Lawson topology on the space of formal balls. In this paper, we characterize spaces Lp(Ω,Σ, μ) on which the hyperbolic topology induced by the norm ∥·∥p coincides with the norm topology. We show the following. (1) The hyperbolic topology and the norm topology coincide for 1 < p < ∞. (2) They coincide on L1(Ω,Σ, μ) if and only if μ(Ω) = 0 or Ω has a finite partition by atoms. (3) They coincide on L∞(Ω,Σ, μ) if and only if μ(Ω) = 0 or there is an atom in Σ.
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