Characterization of Parseval Frame Let $H$ be a Hilbert space, and let $e_j\in H$ for every $j\in\mathbb N$.

Is it possible to show that

$$\left\|f\right\|^2=\sum_{j=1}^\infty\left|\langle f,e_j\rangle\right|^2\text{ for every }f\in H\text{ if and only if }f=\sum_{j=1}^\infty\langle f,e_j\rangle e_j\text{ for every }f\in H\tag*{?}$$

I get the feeling that we need $e_j\perp e_k$ if $j\neq k$.
For the reverse implication see this answer.
Note that orthogonality is not required for a Parseval Frame.
Assume that $\|f\|^2=\sum_j|\langle f,e_j\rangle|^2$ for all $f$. Given $m&gt;n$,
\begin{align}
\Big\|\sum_{j=1}^m\langle f,e_j\rangle\,e_j-\sum_{j=1}^n\langle f,e_j\rangle e_j\Big\|^2
&amp;=\Big\|\sum_{j=n+1}^m\langle f,e_j\rangle e_j\Big\|^2\\[0.3cm]
&amp;=\sup\Big\{\Big|\Big\langle\sum_{j=n+1}^m\langle f,e_j\rangle e_j,h\Big\rangle\Big|:\ \|h\|=1\Big\}\\[0.3cm]
&amp;=\sup\Big\{\Big|\sum_{j=n+1}^m\Big\langle\langle f,e_j\rangle e_j,h\Big\rangle\Big|:\ \|h\|=1\Big\}\\[0.3cm]
&amp;\leq\sup\Big\{\sum_{j=n+1}^m|\langle f,e_j\rangle\,\langle e_j,h\rangle|:\ \|h\|=1\Big\}\\[0.3cm]
&amp;\leq\bigg(\sum_{j=n+1}^m|\langle f,e_j\rangle|^2\bigg)^{1/2}
\sup\Big\{\bigg(\sum_{j=n+1}^m|\langle e_j,h\rangle|^2\bigg)^{1/2}:\ \|h\|=1\Big\}\\[0.3cm]
&amp;\leq\bigg(\sum_{j=n+1}^m|\langle f,e_j\rangle|^2\bigg)^{1/2}
\sup\Big\{\bigg(\sum_{j=1}^\infty|\langle e_j,h\rangle|^2\bigg)^{1/2}:\ \|h\|=1\Big\}\\[0.3cm]
&amp;=\bigg(\sum_{j=n+1}^m|\langle f,e_j\rangle|^2\bigg)^{1/2}.
\end{align}
The convergence of $\sum_j|\langle f,e_j\rangle|^2$ guarantees that the last series can be made arbitrarily small if $m$ and $n$ are large enough. So
$$
\sum_j\langle f,e_j\rangle e_j
$$
exists. Now we need to show that it equals $f$. We will need the Polarization Identity
$$
\langle f,g\rangle=\tfrac14\,\sum_{k=0}^3i^k\|f+i^kg\|^2,
$$
and also the particular case to complex numbers,
$$
z\overline w=\tfrac14\,\sum_{k=0}^3i^k|z+i^kw|^2.
$$
We have
\begin{align}
\langle f,g\rangle&amp;=\tfrac14\,\sum_{k=0}^3i^k\|f+i^kg\|^2
=\tfrac14\,\sum_{k=0}^3i^k\sum_j|\langle f+i^kg,e_j\rangle|^2\\[0.3cm]
&amp;=\sum_j\tfrac14\,\sum_{k=0}^3i^k|\langle f,e_j\rangle+i^k\langle g,e_j\rangle|^2\\[0.3cm]
&amp;=\sum_j\langle f,e_j\rangle\langle e_j,g\rangle.
\end{align}
Then
\begin{align}
\Big\langle f-\sum_j\langle f,e_j\rangle e_j,g\Big\rangle
&amp;=\langle f,g\rangle-\sum_j\langle f,e_j\rangle\langle e_j,g\rangle=0.
\end{align}
As this can be done for any $g$, it follows that
$$
f=\sum_j\langle f,e_j\rangle e_j. 
$$
If $(e_j)$ is a sequence in $H$ with $&lt;e_j,e_k&gt;=\delta_{jk}$, then we have:

$(e_j)$ is an orthonormalbasis of $H$ $ \iff$

$\left\|f\right\|^2=\sum_{j=1}^\infty\left|\langle f,e_j\rangle\right|^2\text{ for every }f\in H \iff\sum_{j=1}^\infty\langle f,e_j\rangle e_j\text{ for every }f\in H\tag*{}$