Smooth points and strictly extreme points in subspaces (Q1345866)

For any bounded point \(x\) of the unit ball in a real normed space, there exists a direct sum decomposition \(X= T(x) \oplus F(x)\) such that \(x\) is both a smooth point of the unit ball in \(T(x) \oplus \text{span} [x]\) and a strictly extreme point of the unit ball in \(F(x)\). Here \(T(x)\) is the tangent space of the unit ball at \(x\), which has important geometric and analytic characteristics. Moreover the notions of quasi-smooth spaces and quasi-strictly convex spaces are introduced and the dual relationship between them is established. As an application, a result of J. Hagler and F. Sullivan is improved.