A normed space $E$ has a 

 -  Kadec property if the norm- and weak topologies coicide on the unit sphere of $E$.

 -  Kadec-Klee property if any sequence on the unit sphere, that is weakly convergent is also norm-convergent (this seems to be the same as Radon-Riesz property, if I am not mistaken).

 -  Dual Kadec (-Klee) property if the norm and weak* topologies (sequence convergences) coincide on the unit sphere of $E^{*}$.

I am not a Banach-space theorist, and so these properties seem somewhat intangible for me, so for writing purposes, I would like to know 
>>more readily verifiable condition on the space $E$, that would imply that $E^{*}$ has a Kadec-Klee property.

I was able to find that any locally uniformly rotund space has a Kadec-Klee property. $E$ is said to be locally uniformly rotund (LUR) if $\lim\|x_n+x_0\|=2\lim x_n=2\|x_0\|$ implies that $x_0=\lim x_n$. If $E^*$ is LUR, then the norm of $E$ is Frechet-differentiable, but the converse seems to fail. A **uniform smoothness** of $E$ implies that $E^*$ is **uniformly rotund**, and so LUR, and hence KK, but this seems to be too restrictive.