I need to prove that if the image of the unit ball $X_1$ of a Banach space $X$ along an operator $A:X\rightarrow Y$ is precompact, $A$ is bounded.
Is my solution correct?
$\|A\|=\sup_{\|x\|=1}\|Ax\|_Y=\sup_{A(X_1)}\|v\|_Y=\sup_{\overline{A(X_1)}}\|v\|_Y=\max_{\overline{A(X_1)}}\|v\|_Y<\infty.$ The third equality is by continuity of the norm and the fourth is by the precompactness of $A(X_1)$.