How can I prove the following? compact operator on a dual space

Question

Let X be Banach space, $ x_{0}\in X $ and $\phi_{0}\in X^{*}$. Define$ T:X^{*}\to X^{*} $ bt $ T(\psi) =\psi (x_{0})\phi_{0}$ for $ \psi\in X^{*} $ then T is compact in $ X^{*} $

Answer 1

The range of $T$ is one-dimensional, it is contained in $span(\phi_0)$. So $T$ is compact.