A study on best approximation and Banach algebra in $n$-normed linear space
Prasenjit Ghosh, Tapas Kumar Samanta
Received December 10, 2024. Published online July 31, 2025.
Abstract: The idea of best approximation in linear $n$-normed space is presented and some examples showing various possibilities of best approximations in linear $n$-normed space is given. Also, we study strictly convex $n$-norm and enquire about the uniqueness of best approximations in $n$-normed linear space. Furthermore, best approximations in $n$-Hilbert space is discussed. Moreover, the notion of a Banach algebra in $n$-Banach space is presented and some examples are discussed. A set-theoretic property of invertible and noninvertible elements in an $n$-Banach algebra is explained and then topological divisor of zero in $n$-Banach algebra is defined. Finally, we introduce the notion of a complex homeomorphism in an $n$-Banach algebra and derive Gleason, Kahane, Zelazko type theorem with the help of complex $b$-homeomorphism in the case of $n$-Banach algebra.
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Affiliations: Prasenjit Ghosh (corresponding author), Department of Mathematics, Barwan N. S. High School (HS), Barwan, Murshidabad, 742161, West Bengal, India; e-mail: prasenjitpuremath@gmail.com; Tapas Kumar Samanta, Department of Mathematics, Uluberia College, Uluberia, Howrah, 711315, West Bengal, India, e-mail: mumpu_tapas5@yahoo.co.in
