On Hahn-Banach smoothness and related properties in Banach spaces

Abstract

In this paper, we study several variants of Hahn-Banach smoothness, viz., property-$(SU)$/$(HB)$/$(wU)$, where property-$(SU)$ and property-$(HB)$ are stronger notions and property-$(wU)$ is a weaker notion of Hahn-Banach smoothness. We characterize property-$(wU)$ and property-$(HB)$. It is observed that $L_1(\mu )$ has property-$(wU)$ in $L_1(\mu ,({\mathbb{R}}^2,\Vert .{\Vert }_2))$ but it does not have property-$(U)$ in $L_1(\mu ,({\mathbb{R}}^2,\Vert .{\Vert }_2))$ for a non-atomic measure $\mu$. We derive a sufficient condition when property-$(wU)$ is equivalent to property-$(U)$ of a subspace. It is observed that these properties are separably determined. Finally, finite-dimensional and finite co-dimensional subspaces of $c_0$, ${\ell }_p$ ($1\le p<\infty$) having these properties are characterized.