Abstract
In this paper, we -orthogonality and explore -symmetricity such as a -left-symmetric (-right-symmetric) of a vector x in a real Banach space (\U0001d54f, ‖·‖\U0001d569) and study the relation between a -right-symmetric (-left-symmetric) in ℐ(x). New results and proofs are include the notion of norm attainment set of a continuous linear functionals on a reflexive and strictly convex Banach space and using these results to characterize a smoothness of a vector in a unit sphere.
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