On the continuity properties of the Lp balls

Abstract

Abstract In this paper the right upper semicontinuity at p = 1 {p=1} and continuity at p = ∞ {p=\infty} of the set-valued map p → B Ω , 𝒳 , p ⁢ ( r ) {p\rightarrow B_{\Omega,\mathcal{X},p}(r)} , p ∈ [ 1 , ∞ ] {p\in[1,\infty]} , are studied where B Ω , 𝒳 , p ⁢ ( r ) {B_{\Omega,\mathcal{X},p}(r)} is the closed ball of the space L p ⁢ ( Ω , Σ , μ ; 𝒳 ) {L_{p}(\Omega,\Sigma,\mu;\mathcal{X})} centered at the origin with radius r, ( Ω , Σ , μ ) {(\Omega,\Sigma,\mu)} is a finite and positive measure space, 𝒳 {\mathcal{X}} is a separable Banach space. It is proved that the considered set-valued map is right upper semicontinuous at p = 1 {p=1} and continuous at p = ∞ {p=\infty} . 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.