Lemma 37.3.6 (Amplification).label Let $H$ be a complex Hilbert space, $n \in \natp$, and
\[\pi: B(H) \to B(H^{n}) \quad [\pi(T)(x)]_{n} = Tx_{n}\]
then for each $\seqf{x_j}\subset H$,
\[\max_{1 \le j \le n}\norm{Tx_j}_{H} \le \norm{\pi(T)(x)}_{H^n}\le n \max_{1 \le j \le n}\norm{Tx_j}_{H}\]
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