标题： 局部凸空间中Hahn-Banach延拓的唯一性 显示英文标题

标题： Uniqueness of Hahn-Banach extensions in locally convex spaces

摘要： 我们打算研究局部凸空间中子空间上的线性泛函的Hahn-Banach延拓的唯一性。当子空间 $Y$ 在局部凸空间中具有Phelps引入的类似性质-U（称为性质-SNP）时，推导出各种特征化。我们刻画了所有子空间都具有此性质的空间。证明表明，Banach空间 $E$ 的子空间 $M$ 具有性质-U 当且仅当该局部凸空间 $E$ 赋予弱拓扑后的子空间 $M$ 具有上述提到的性质-SNP。这项研究围绕探索半范族与之前提到的泛函唯一延拓之间的潜在联系展开。我们广泛研究了Tychonoff空间上赋予逐点收敛拓扑的连续函数空间中的这一性质。 显示英文摘要

摘要： We intend to study the uniqueness of the Hahn-Banach extensions of linear functionals on a subspace in locally convex spaces. Various characterizations are derived when a subspace $Y$ has an analogous version of property-U (introduced by Phelps) in a locally convex space, referred to as the property-SNP. We characterize spaces where every subspace has this property. It is demonstrated that a subspace $M$ of a Banach space $E$ has property-U if and only if the subspace $M$ of the locally convex space $E$ endowed with the weak topology has the property-SNP, mentioned above. This investigation circles around exploring the potential connections between the family of seminorms and the unique extension of functionals previously mentioned. We extensively studied this property on the spaces of continuous functions on Tychonoff spaces endowed with the topology of pointwise convergence.