Hyperbolic topology of normed linear spaces

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In a previous paper [H. Tsuiki, Y. Hattori, Lawson topology of the space of formal balls and the hyperbolic topology of a metric space, Theoret. Comput. Sci. 405 (2008) 198–205], 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 L p ( 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 L 1 ( 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.