Let $X, Y$ be infinite dimensional, Banach spaces. Let $\mathcal{ L}(X, Y)$ be the space of bounded operators . Motivated by the fact that smoothness of norm in the higher duals of even order of a Banach space can lead to Frechet differentiability, we exhibit classes of Banach spaces $X, Y$ where very smooth points (i.e., smooth points that remain smooth in the bidual) in the space of compact operators $\mathcal{ K}(X, Y)$ are Frechet smooth in $\mathcal{ L}(X, Y)$ and hence in $\mathcal{ K}(X, Y)$.
Categories: math.FA