Let $\left\{e_k\right\}_{k=1}^{\infty}$ be an an orthonormal basis for the Hilbert space $\mathcal H$.
Prove that, if $\sum_{k=1}^{\infty}\left|\left\langle f, e_k\right\rangle\right|^2=\|f\|^2$, $\forall f \in \mathcal{H}$ then $\overline{\operatorname{span}}\left\{e_k\right\}_{k=1}^{\infty}=\mathcal{H}$.
I don't know how to start with this problem. I know that if $\left\{e_k\right\}_{k=1}^{\infty}$ is an orthonormal basis then $f=\sum_{k=1}^{\infty}\left\langle f, e_k\right\rangle e_k, \forall f \in \mathcal{H}$ and so $\langle f, g\rangle=\sum_{k=1}^{\infty}\left\langle f, e_k\right\rangle\left\langle e_k, g\right\rangle, \forall f, g \in \mathcal{H}$. From this we have that $\sum_{k=1}^{\infty}\left|\left\langle f, e_k\right\rangle\right|^2=\|f\|^2, \forall f \in \mathcal{H}$. But how can I infer that $\overline{\operatorname{span}}\left\{e_k\right\}_{k=1}^{\infty}=\mathcal{H}$?
Best Answer
The following is an orthogonal decomposition $$ f=\left(f-\sum_{k=1}^{n}\langle f,e_{k}\rangle e_k\right)+\sum_{k=1}^{n}\langle f,e_{k}\rangle e_{k}. $$ Therefore, $$ \|f\|^2=\left\|f-\sum_{k=1}^{n}\langle f,e_k\rangle e_k \right\|^2+\sum_{k=1}^{n}|\langle f,e_{k}\rangle|^2. $$ I'll let you take it from there.