I have to prove that the operator $T\colon\ell_\infty\to\mathcal{L}(\ell_2,\ell_2)$ such $T((a_j)_{j=1}^{\infty})((b_j)_{j=1}^{\infty})=(a_jb_j)_{j=1}^{\infty}$ is an linear isometry.
How can I show that $T$ is an isometry, $( ||T(b_n||)=||b_n|| )$
can I use this to show that $\mathcal{L}({\ell_2,\ell_2})$ is not separable? I know that every normed separable space is isometrically isomorphic to a subspace of $\ell_\infty$ so I have to prove that $T$ is not a bijection. Is this true?