Let $H$ be any Hilbert space. How can we prove that any bounded linear operator $T\colon H \to \ell^1$ is compact?
If we use the fact that the space $\ell^1$ has Schur property (norm and weak convergence is the same), then we need to show that for a sequence $(x_n) \subset H$ such that $||x_n|| \leq 1$ the sequence $(Tx_n) \subset \ell^1$ contains weakly convergent subsequence. But I do not know how I should do this.
What property of a Hilbert spaces I need to use?