On a new space consisting of norm maintaining functions

2019/12/03 by Manuel Norman, Norman, Manuel
Mathematics · Computer Science · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.1912.01375

Abstract

In this paper we define a new space, LH(X,Y), consisting of functions f∈ X ⊂ Y (with X,Y normed spaces) such that ‖ f ‖_X ≡ ‖ f ‖_Y (where ‖ ⋅ ‖_X is any norm on X, in general not the norm induced by ‖ ⋅ ‖_Y on X). In Section 2 we study some properties involving LH and Schur's property, norm attainment and LHW (a weaker version of LH). In particular, one of the main results of this section states that strong and weak norm attainments together with some conditions imply that the considered function belongs to LHW or LH (depending on which conditions are satisfied). In Section 3 we renorm in a natural way the space Y so that LH(X,\widetildeY)=X, obtaining an important extension Theorem.

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