[[Orthonormal dense basis]]
# Parseval's relation

Let $X$ be a [[Hilbert space]] and $\mathcal{E}= \{ \ket{e_{i}} \}_{i=1}^\infty$ be an orthonormal dense basis.
Then for any $\ket{x},\ket{y} \in X$ we have #m/thm/anal/fun 
$$
\begin{align*}
\braket{ x | y } = \sum_{j=1}^\infty \braket{ x | e_{j} } \braket{ e_{j} | y } 
\end{align*}
$$

> [!missing]- Proof
> #missing/proof


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