A Banach space $X$ is called a Grothendieck space if $weak^{*}$-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, to construct a $weak^{*}$-null sequence $(\mu_{n})_{n}$ in $C[0,1]^{*}$ which is not weakly null.

Thank you.

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.