Show that $L$ is compact

Question

Let E and F be Banach spaces; $L:E\rightarrow F$ a linear and continuous application that transforms closed and bounded sets into compact. Show that $L$ is compact.

I have really tried to solve this by my own, but I just cannot figure it out. So please would anybody be kind enough to help me. Thanks in advance.

Answer 1

Isn't that $L(\overline{B_{X}})$ being relatively compact (in fact, it is compact by the assumption)? Here $B_{X}$ is the closed unit ball in $X$. Then the compactness of the operator follows immediately by definition.