| Summary: | 碩士 === 國立中山大學 === 應用數學系研究所 === 94 === Let X be a Banach space and Y be a closed subspace of X. Given a
bounded linear functional f on Y , the Hahn-Banach theorem guarantees
that there exists a linear extension ˜ f 2 X of f which preserves the norm
of f. But it does not state that such ˜ f is unique or not. If every f in Y
does have a unique norm preserving extension ˜ f in X , we say that Y has
the unique extension property, or, following P. R. Phelps, the property U in
X.
A. E. Taylor [17] and S. R. Foguel [7] had shown that every subspace Y
of X has the unique norm-preserving extension property in X if and only if
the dual space X is strictly convex. As known in [11], X is smooth if X is
strictly convex. The converse does not hold in general unless X is reflexive.
In this thesis, we show that if a subspace Y of a Banach space X has
the unique extension property then the norm of Y is outward smooth in X.
The converse holds when Y is reflexive. Note that our conditions are local,
i.e., they depend on Y only, but not on X. Several related results are also
derived. Our work extends and unifies recent results in literature.
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