Multi-smooth points of finite order

A point x of the unit sphere S(X) of the Banach space X is called a multi-smooth point of order n if there exist exactly n-independent continuous linear functionals g₁,⋯, g_n, in S(X*), the unit sphere of the dual of X, such that g_i(x) = 1, for 1 ≤ i ≤ n. The object of this paper is to characterize multi-smooth points of some function and operator spaces.