A study on Best Approximation and Banach Algebra in n-normed linear space

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 non-invertible elements in a 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 a n-Banach algebra and derive Gleason, Kahane, Zelazko type theorem with the help of complex b-homeomorphism in the case of n-Banach algebra.