Parseval identity for a Hilbertian basis

For a Hilbertian basis

(e_{i})

,

⟨ x, y ⟩ = \sum_{i} ⟨ x, e_{i} ⟩ \overline{⟨ y, e_{i} ⟩}, ∥ x ∥^{2} = \sum_{i} ∣ ⟨ x, e_{i} ⟩ ∣^{2} .

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