Parseval's Identity

functional-analysis

Definition

Parseval's Identity

Let $\mathcal{H}$ be a Hilbert space with orthonormal basis $\{e_i{\}}_{i\in I}$. For any $v\in \mathcal{H}$ with expansion $v={\sum }_i{c}_i{e}_i$ where $c_i=⟨e_i|v⟩$:

$$\Vert v{\Vert }^2=\sum _{i\in I}|c_i{|}^2=\sum _{i\in I}|⟨e_i|v⟩{|}^2$$

Equivalently:

$$⟨v|v⟩=\sum _{i\in I}⟨v|e_i⟩⟨e_i|v⟩$$

The identity expresses that the inner product is preserved under the isomorphism $\mathcal{H}\cong {\ell }^2(I)$.