On strong orthogonality and strictly convex normed linear spaces

Abstract

We introduce the notion of a 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 [InlineEquation not available: see fulltext.] to be an exposed point of the unit ball [InlineEquation not available: see fulltext.]. 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 λx with [InlineEquation not available: see fulltext.]. MSC: 46B20, 47A30. © 2013 Paul et al.; licensee Springer.