A direct example on $X^*$ is reflexive but the weak topology and weak* topology on $X^*$ are not equal for a normed space $X$.

Question

I asked this question Example on $X^*$ is reflexive but $\sigma(X^*,X)\neq\sigma(X^*,X^{**})$ for a normed space $X$. before. And the answer is to use $X$ is reflexive iff $\sigma(X^*,X)=\sigma(X^*,X^{**})$ to find a normed space X is not reflexive but $X^*$ is reflexive.

But my professor told me that there is an easy example that show $X^*$ is reflexive but $\sigma(X^*,X)\neq\sigma(X^*,X^{**})$ without using the equivalence. Can anyone give me a hint? Thank you in advance!