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Orthogonality and smoothness induced by the norm derivatives

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

We study the concepts of orthogonality and smoothness in normed linear spaces, induced by the derivatives of the norm function. We obtain analytic characterizations of the said orthogonality relations in terms of support functionals in the dual space. We also characterize the related notions of local smoothness and establish its connection with the corresponding orthogonality set, which is analogous to the well-known relation between the Birkhoff–James orthogonality and the classical notion of smoothness. The similarities and the differences between the various notions of smoothness are illustrated by considering some particular examples, including \( {\mathbb {K}}({\mathbb {H}}), \) the Banach space of all compact operators on a Hilbert space \( {\mathbb {H}}. \)

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  1. Department of Mathematics, Indian Institute of Science, Bengaluru, 560012, Karnataka, India

    Debmalya Sain

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  1. Debmalya Sain
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The author feels elated to acknowledge the delightful friendship of Miss Irin Parvin, while fondly remembering the beautiful times created together.

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Sain, D. Orthogonality and smoothness induced by the norm derivatives. RACSAM 115, 120 (2021). https://doi.org/10.1007/s13398-021-01060-0

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