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arXiv · 2311.00942

Directional Differentiability of the Metric Projection in Bochner Spaces

Jinlu Li
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Abstract

In this paper, we consider the directional differentiability of metric projection and its properties in uniformly convex and uniformly smooth Bochner space Lp(S; X), in which (S, A, mu) is a positive measure space and X is a uniformly convex and uniformly smooth Banach space. Let (arbitrary) A in A with measure of A greater than 0 and define a subspace Lp(A; X) of Lp(S; X), which is considered as a closed and convex subset of Lp(S; X). We first study the properties of the normalized duality mapping in Lp(S; X) and in Lp(A; X). For any c in Lp(A; X) and r > 0, we define a closed ball BA(c; r) in Lp(A; X) and a cylinder CA(c; r) in Lp(S; X) with base BA(c; r). Then, we investigate some optimal properties of the corresponding metric projections P(Lp(A;X)), P(BA(c;r)) and P(CA(c;r)) that include the inverse images, the directional differentiability and the precise solutions of their directional derivatives.

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Jinlu Li. 2023-11-02. Directional Differentiability of the Metric Projection in Bochner Spaces. https://arxiv.org/abs/2311.00942

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