TAYLOR-FOUGEL定理的一个推广

A Generalization of Taylor—Fougel Theorem

在本文中,我们证明Banach空间X的每个子空间皆有(U_k)性质,当且仅当X~*为k-严格凸的,这是Taylor-Fougel定理的推广。

Let X be a Banach space and M a subspace of X. For every f∈M~*, set E_M(f)={g∈X~*: ‖g‖=‖f‖_M and g(x)=f(x) for x∈M}. The subspace M is said to have the property (U_k) if for every f∈M~*, the dimE_M(f) satisfies the inequality dimE_M(f)≤k, where k is a positive integral. In this paper, the following results are obtained. Theorem 1. Every subspace of a Banach space X has property (U_k) if and only if X~* is k-strictly convex. This is a generalization of Taylor-Fougel theorem. Theorem 2. If X is k-UR space, then every subspace of X~* has property