On strong orthogonality and strictly convex normed linear spaces

Abstract

We introduce the notion of strongly orthogonal set relative to an element in the sense of Birkhoff-James in a normed linear space to find a necessary and sufficient condition for an element $x$ of the unit sphere $S_X$ to be an exposed point of the unit ball $B_X.$ We then prove that a normed linear space is strictly convex iff for each element x of the unit sphere there exists a bounded linear operator A on X which attains its norm only at the points of the form $\hat{I}»x$ with $\hat{I}»\in S_K$.