On the norm attainment set of a bounded linear operator and semi-inner-products in normed spaces

TL;DR

This paper characterizes the norm attainment set of bounded linear functionals and operators in normed spaces using semi-inner-products, providing new insights into the geometry of these spaces.

Contribution

It offers a complete characterization of norm attainment sets via semi-inner-products, answering an open question and enhancing understanding of normed space geometry.

Findings

Complete characterization of norm attainment set for functionals.

Extension of characterization to bounded linear operators.

Application of semi-inner-products to geometric analysis of normed spaces.

Abstract

We obtain a complete characterization of the norm attainment set of a bounded linear functional on a normed space, in terms of a semi-inner-product defined on the space. Motivated by this result, we further apply the concept of semi-inner-products to obtain a complete characterization of the norm attainment set of a bounded linear operator between any two real normed spaces. In particular, this answers an open question raised recently in [Sain, D., \textit{On the norm attainment set of a bounded linear operator}, J. Math. Anal. Appl., \textbf{457} (2018), 67-76.]. Our results illustrate the applicability of semi-inner-products towards a better understanding of the geometry of normed spaces.