In this paper, we study two properties viz., property-U and property-SU of a subspace Y of a Banach space X, which correspond to the uniqueness of the Hahn–Banach extension of each linear functional in $Y^{\ast }$ and when this association forms a linear operator of norm-1 from $Y^{\ast }$ to $X^{\ast }$. It is proved that, under certain geometric assumptions on X, Y, Z, these properties are stable with respect to the injective tensor product; Y has property-U (SU) in Z if and only if $X{\otimes }_{\epsilon }^{\vee }Y$ has property-U (SU) in $X{\otimes }_{\epsilon }^{\vee }Z$. We prove that when $X^{\ast }$ has the Radon–Nikodým Property for 1 < p < ∞, L_p(μ, Y) has property-U (property-SU) in L_p(μ, X) if and only if Y is so in X. We show that if Z⊆ Y⊆ X and Y has property-U (SU) in X then Y/Z has property-U (SU) in X/Z. On the other hand, Y has property-SU in X if Y/Z has property-SU in X/Z and Z (⊆ Y) is an M-ideal in X. This partly solves the 3-space problem for property-SU. We characterize all hyperplanes in c₀ which have property-SU. We derive necessary and sufficient conditions for all finite codimensional proximinal subspaces of c₀ which have property-U (SU).

在本文中，我们研究了Banach 空间X的子空间Y的两个属性，即属性 - U和属性 - SU，它们对应于每个线性泛函的 Hahn-Banach 扩展的唯一性${是}^{＊}$当此关联形成 norm-1 的线性算子时${是}^{＊}$到$X^{＊}$. 证明了，在X、Y、Z的特定几何假设下，这些性质对于单射张量积是稳定的；Y在Z中具有属性U ( SU )当且仅当$X{\otimes }_{\epsilon }^{\vee }是$有财产- U ( SU ) 在$X{\otimes }_{\epsilon }^{\vee }Z$. 我们证明当$X^{＊}$对于 1 < p  < ∞具有 Radon–Nikodým 特性 ， L _p ( μ , Y )在L _p ( μ , X ) 中具有特性- U (特性- SU )当且仅当Y在X中也是如此。我们证明如果Z ⊆ Y ⊆ X并且Y在X中具有属性- U ( SU )那么Y / Z在X /中具有属性 - U ( SU )Z。 _ 另一方面，如果Y / Z在X / Z中具有属性- SU且Z (⊆ Y ) 是 X中的 M-理想，则 Y 在X中具有属性 - SU 。这部分解决了属性SU的 3 空间问题。我们刻画了c ₀中所有具有属性SU的超平面。我们推导出c ₀的所有有限余维临近子空间的充分必要条件，这些子空间具有属性- U ( SU )。