Let $\mathcal{H}$ be a Hilbert space and let $\mathcal{K}$ be a Hilbert space with an orthonormal basis $\{ e_i \}_{i \in I}$. Let $A$ be bounded linear operator from $\mathcal{H} \otimes \mathcal{K}$ to $\mathcal{H} \otimes \mathcal{K}$. How to show that $\| A \|^2 \leq \sum_{i,j} \|(I_{\mathcal{H}} \otimes \left< e_i \right| ) A (I_{\mathcal{H}} \otimes \left| e_j \right>)\|^2,$
where $\left< \cdot \right|$ and $\left| \cdot \right>$ are Dirac "bra" and "ket"?
Is it true that if $A$ is a bounded linear operator on a Hilbert space $\mathcal{H}$ then $\|A\| = \sup_{N}\|P_{N}A\|?$ The operator $P$ is a projection onto $N \subseteq \mathcal{H}$ and the supremum is taken over all finite-dimensional subspaces of $\mathcal{H}$.
Thank you for the help.