Manuscript En07-C@1 · En07-C.tex · lemma · 3f7e9ebb

Lemma: Finite-code strong convergence

Access-preserving endpoint certificates › Endpoint topology and finite autonomous evolution · lines 1705–1721 · proof included or linked

En07-C@1#lem:strong

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For a fixed finite-dimensional code $C$ , pointwise norm convergence of $V_{t} ∣ ψ ⟩$ to $V_{*} ∣ ψ ⟩$ for every $∣ ψ ⟩ \in C$ implies operator-norm convergence.

Proof. For an orthonormal basis ${e_{i}}_{i = 1}^{d}$ and $D_{t} = V_{t} - V_{*}$ ,

∥ D_{t} ∥_{\infty} \leq (i = 1 \sum^{d} ∥ D_{t} e_{i} ∥^{2})^{1/2} .

Each summand tends to zero.

Displayed formulas in this passage

unlabeled · line 1715
$∥ D_{t} ∥_{\infty} \leq (i = 1 \sum^{d} ∥ D_{t} e_{i} ∥^{2})^{1/2} .$

Nearby in this section

section: Endpoint topology and finite autonomous evolution
proposition: Proposition: A fixed autonomous unitary does not supply a nontrivial norm endpoint
section: Endpoint topology and finite autonomous evolution