On strong orthogonality and strictly convex normed linear spaces

TL;DR

This paper introduces a new concept of strong orthogonality in normed linear spaces to characterize exposed points of the unit ball and establishes a link between strict convexity and the existence of specific norm-attaining operators.

Contribution

It defines strongly orthogonal sets relative to an element and provides a characterization of strict convexity via norm-attaining operators.

Findings

Strong orthogonality characterizes exposed points of the unit ball.

Strict convexity is equivalent to the existence of operators attaining their norm only at scalar multiples of a given element.

The paper establishes necessary and sufficient conditions linking geometry and operator theory in normed 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 $ \lambda x $ with $ \lambda \in S_{K} $.