1.19 Parseval's and plancherel's theorems

When dealing with transform coefficients, we will see that our notions of distance and angle carry over to the coefficient space.

Let $x,y\in V$ and suppose that ${\left\{v_k\right\}}_{k\in \Gamma }$ is an orthobasis. ( $\Gamma$ denotes the index set, which could be finite or infinite.) Then $x={\sum }_{k\in \Gamma }{\alpha }_k{v}_k$ and $y={\sum }_{k\in \Gamma }{\beta }_k{v}_k$ , and

So

This is Plancherel's theorem. Parseval's theorem follows since ${〈x,,,x〉}_V={〈\alpha ,,,\alpha 〉}_{{\ell }_2}$ which implies that ${∥x∥}_V^2={∥x∥}_{{\ell }_2}^2$ . Thus, an orthobasis makes every inner product space equivalent to ${\ell }_2$ !