Parseval’s relation

Let $𝑋$ be a Hilbert space and $E = {| 𝑒 𝑖 ⟩} \infty 𝑖 = 1$ be an orthonormal dense basis. Then for any $| 𝑥 ⟩, | 𝑦 ⟩ \in 𝑋$ we have fun

⟨ 𝑥 | 𝑦 ⟩ = \infty \sum 𝑗 = 1 ⟨ 𝑥 | 𝑒 𝑗 ⟩ ⟨ 𝑒 𝑗 | 𝑦 ⟩

Proof

proof