Gâteaux differentiable points with special representation

Abstract:

If $({x_n})$ is a bounded sequence in a Banach space, is there an element $x = \sum \nolimits _{n = 1}^\infty {{a_n}{x_n}}$ sucn that $\sum \nolimits _{n = 1}^\infty {\left \| {{a_n}{x_n}} \right \| < \infty }$ and tne directional derivative of the norm at $x$, $D(x,{x_n})$, exists for every $n$? In fact, there are such $x$’s dense in the closed span of $\left \{ {{x_n}} \right \}$. An application of this fact is made to a proof of Rybakov’s theorem on vector measures.