On the continuity properties of the Lp balls

Abstract

In this paper the right upper semicontinuity at p = 1 and continuity at p = ∞ of the set-valued map p → B_{\Omega,X,p}(r), p ∈ [1,∞], are studied where B_{\Omega,,X,p}(r) is the closed ball of the space L_p(\Omega, Σ, \mu;X) centered at the origin with radius r, (\Omega, Σ,\mu) is a finite and positive measure space, X is a separable Banach space. It is proved that the considered set-valued map is right upper semicontinuous at p = 1 and continuous at p = ∞. An application of the obtained results to the set of integrable outputs of the input-output system described by the Urysohn-type integral operator is discussed.