## A Generalization of Taylor—Fougel Theorem

**Nan Chaoxun(Department of Mathematics)**

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

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