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We obtain a complete characterization of the norm attainment set of a bounded linear operator between normed spaces, in terms of semi-inner-product(s) defined on the space. In particular, this answers an open question raised recently in [D. Sain, On the norm attainment set of a bounded linear operator, J. Math. Anal. Appl., 457 (2018), 67–76]. Our results illustrate the applicability of semi-inner-products towards a better understanding of the geometry of normed spaces.

On the Norm Attainment Set of a Bounded Linear Operator and Semi-Inner-Products in Normed Spaces