Let $X,Y$ be Banach spaces. $T\colon X\to Y$ be a bounded linear operator.
How can I prove that $T$ is compact if and only if there is $\lbrace x_n^*\rbrace\subset X^*$ such that $\|x_n^*\|\to 0$ and $\|T(x)\|\leq \operatorname{sup}_n|x_n^*(x)|$ for every $x\in X$?