On the continuity properties of the Lp balls

In this paper the right upper semicontinuity at p = 1 and continuity at p = ∞ of the set-valued map p → BΩ,X,p(r), p ∈ [1, ∞], are studied where BΩ,X,p(r) is the closed ball of the space Lp(Ω, Σ, μ;X) centered at the origin with radius r, (Ω, Σ, μ) 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.

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Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).