  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.
