Smoothness in some Banach Spaces of Operators and Vector Valued Functions

A well known criterion of \v{S}mulyan states that the norm $\|\cdot\|$ of a real Banach space $X$ is G\^{a}teaux differentiable at $x\in X$ if and only if there is $x^*\in S_{X^*}$ which is $w^*$-exposed by $x$ in $B_{X^*}$ and that the norm is Fr\'echet differentiable at $x$ if and only if there is $x^*\in S_{X^*}$ which is $w^*$-strongly exposed in $B_{X^*}$ by $x$. We show that in this criterion $B_{X^*}$ can be replaced by a convenient smaller set, and we apply this extended criterion to characterize the points of G\^{a}teaux and Fr\'echet differentiability of the norm in epsilon products of Banach spaces, extending previous work of Heinrich. As a consequence we get some results of smoothness of the norm in some Banach spaces of continuous and harmonic vector valued functions.