On a new space consisting of norm maintaining functions

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.