令X 為Banach 空間，Y 為其一個閉子空間。給定一個Y 上的有界線性泛函f，根據Hahn-Banach 定理，我們知道在X 上存在一個有界線性泛函g 為f 的擴張並且保持相同範數。但這樣的一個擴張函數並不是唯一的。如果Y 上的每一個有界線性泛函都有如上述的唯一擴張，我們就稱Y 具有唯一擴張性，或者引述P. R. Phelps 的說法，稱作Y 在X 上有property U。而A. E. Taylor和S. R. Foguel分別在1939 與1958 年證明了Y有唯一擴張性的等價條件為X的對偶空間X*是嚴格凸(strictly convex)。並且我們知道如果空間X的對偶空間X*是嚴格凸，則原空間X是平滑的(smooth)，然而此敘述的逆向敘述必須在X為自反(reflexive)的情況下才成立。在此篇論文中，我們證明了如果一個Banach 空間X 的閉子空間Y 有唯一擴張性，則Y 的範數在X 上為外向光滑(outward smooth)；當Y 為自反的時候逆向敘述會成立。

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.

Contents
1 Introduction 4
2 Preliminaries and notations 7
3 Main results 13
References 18

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