A Banach space $X$ is called a Grothendieck space if $\text{weak}^{\*}$-null sequences in $X^{\*}$ are weakly null. Some of the classical Grothendieck spaces are the $C(\Omega)$ spaces if $\Omega$ is extremally disconnected. However, $C[0,1]$ is not a Grothendieck space. This fact can easily be deduced from some well-known characterizations of Grothendieck spaces, such as, $C(\Omega)$ is a Grothendieck space if and only if it does not contain any complemented copy of $c\_{0}$. But my concern is whether there is a direct proof of this fact, that is, a construction of a $\text{weak}^{\*}$-null sequence $(\mu\_{n})\_{n}$ in $C[0,1]^{\*}$ which is not weakly null.



$C[0,1]$ is not a Grothendieck space
Consider $\delta\_{1/n}-\delta\_0$; this defines a weak$^\*$ null sequences which is not weakly null; e.g., $\langle \delta\_{1/n}-\delta\_0, \chi\_{\{0\}} \rangle \not\to 0$. So if $\Omega$ contains a nontrivial convergent sequence, $C(\Omega)$ cannot be a Grothendieck space.


