Is the following definition of orthonormal basis in Hilbert space?

Question

Let $H$ Hilbert, then $\{e_n \} \subset H$ is an orthonormal basis of $H$ if $|e_n|=1$ for every $n$, $\langle e_n,e_m \rangle =0$ for every $m,n$ and $\overline {\operatorname{span}(\{e_n \})}=H$.

Recall that $\operatorname{span}(\{e_n \})$ is the space of finite linear combinations.

Now call $\infty-\operatorname{span}(\{e_n \})$ the space of linear combinations of infinitely many elements.

Is it the same in the definition of orthonormal basis to ask that $\infty-\operatorname{span}(\{e_n \})=H$ (i.e. like it happens for abstract Fourier series)?

Answer 1

The condition $\overline {\operatorname{span}(\{e_n \})}=H$ is equivalent to $$ {\operatorname{span}(\{e_n \})}^\perp=\{0\}. $$ This is enough to show that every $h\in H$, $$ h=\sum_n\langle h,e_n\rangle\,e_n. $$