Uniqueness of Hahn-Banach extensions in locally convex spaces
Sainik Karak, Akshay Kumar, Tanmoy Paul
TL;DR
The paper addresses the problem of uniqueness of Hahn-Banach extensions in locally convex spaces by introducing the seminorm-preserving properties SNP and USNP, and linking their validity to the Haar property of annihilators in the dual space. It characterizes when subspaces have SNP (equivalently, when $Y^{\perp}$ has the Haar property in $X^*$) and shows that the all-subspace condition is equivalent to every line in $X^*$ having the Haar property, extending classical Banach-space results to the LCS setting. It further develops the $Y_{lcs}^{\#}$ framework to decompose functionals and to analyze how extension properties interact with quotient spaces, including instances where SNP fails to pass to quotients. The work culminates with illustrative examples in $C_p(Z)$ spaces and other LCS constructions, illuminating when SNP/USNP hold or fail and clarifying the role of the generating seminorm family in extension uniqueness and representation of dual elements.
Abstract
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.