Norm-Preserving Extensions and Best Approximations

In this paper, the relationship between best approximations and norm-preserving extensions of a linear functional is considered. By introducing the notion "property (k − U)," the following results are obtained: Let M be a subspace of a normed linear space X. Then M has property (k − U) in X if and only if its annihilator M⊥ is a k-Chebyshev subspace of X*. If M is a closed subspace of a reflexive space X, then M is a k-Chebyshev subspace of X if and only if M(⊥) has property (k − U) in X*. Using the notions of property (k − U) and k-Chebyshev subspace, some characterizations of k-strict convexity and k-smoothness are given. Some results in R. R. Phelps [Trans. Amer. Math. Soc. 95 (1960), 238-255] and R. B. Holmes ["Geometric Functional Analysis," Springer-Verlag, New York, 1975] are therefore generalized.