I have the suspect that the following statement is true, but I don't how to prove it. Any suggestion? Thanks to all!
Let $X$, $Y$ be Hilbert spaces and let $T \colon X \to Y$ be a linear continuous injective map. Suppose that for every $\epsilon > 0$ there exists a closed vector subspace $V_\epsilon \subseteq X$ of finite codimension such that $\Vert Tv \Vert_Y \leq \epsilon \Vert v \Vert_X$ for all $v \in V_\epsilon$. Then $T$ is compact.