Prove that an orthonormal set $\{y_1,y_2,y_3,\dots\}$ is an orthonormal basis iff $\sum_{m=1}^\infty \vert(y_m,x_n)\vert^2=1\forall n\in \mathbb N$

Question

Let $\{x_1,x_2,x_3,\dots\}$ be an orthonormal basis for the Hilbert Space $H$ over the field $\mathbb R$ with inner product $(\;\;,\;\;)$. Then, prove that an orthonormal set $\{y_1,y_2,y_3,\dots\}$ in $H$ is an orthonormal basis of $H$ iff $$\sum_{m=1}^\infty \vert(y_m,x_n)\vert^2=1$$ for each $n\in \mathbb N$.

Assuming that $\{y_1,y_2,y_3,\dots\}$ is an orthonormal basis, I can arrive at the required result using Parseval's Identity.

But, I can't show the reverse implication.

Answer 1

The assumptions imply that $$x_n=\sum_m\langle x_n,y_m\rangle y_m$$ Therefore $x_n $ belongs to the closed linear span of the elements $y_m.$ Thus the closed linear span of $\{x_n\}$ is contained in the closed linear span of $\{y_m\}.$