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Uniform basis-tail criterion for strong convergence

Suppose

x_{n} ⇀ x

in a separable Hilbert space with Hilbertian basis

(e_{i})

. Then

x_{n} \to x

in norm exactly when

\forall ϵ > 0 \exists I \forall n, \sum_{i \geq I} ∣ ⟨ x_{n}, e_{i} ⟩ ∣^{2} < ϵ .

The condition prevents norm from escaping to successively higher basis coordinates.

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