On a new space consisting of norm maintaining functions

TL;DR

This paper introduces a new space of functions with equal norms in different spaces, explores its properties, and provides a renorming technique that extends the space to include the original space as a special case.

Contribution

It defines the space $LH(X,Y)$, studies its properties, and presents a renorming method that extends the space to recover the original space $X$.

Findings

Characterization of $LH$ and $LHW$ spaces.

Conditions for norm attainment imply membership in $LHW$ or $LH$.

A renorming technique that extends $Y$ so that $LH(X,\widetilde{Y})=X$.

Abstract

In this paper we define a new space, $LH(X,Y)$, consisting of functions $f\in X \subset Y$ (with $X,Y$ normed spaces) such that $\| f \|_X \equiv \| f \|_Y$ (where $\| \cdot \|_X$ is any norm on $X$, in general not the norm induced by $\| \cdot \|_Y$ 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 $LH(X,\widetilde{Y})=X$, obtaining an important extension Theorem.