Theorem about orthogonal system in inner product space. It is known that "If $\{x_n\}$ is a sequence in a real Hilbert space $H$
satisfying 
$$
\langle x_n, x_m\rangle =0 \quad\forall n\ne m,
$$ then $\displaystyle\sum_{n=1}^{\infty}x_n$ is convergent if and only if $\displaystyle\sum_{n=1}^{\infty}\|x_n\|^2$ is convergent". I would like to know if $H$ is an inner product space (pre-Hilbert space) the above statement is still true?

If $\{x_n\}$ is an orthonormal system then $\|x_n\|=1$ for all $n$, so that $\sum\|x_n\|$ does not converge.
I think that what you want is the following result:
Let $X$ be a normed space with norm $\|\,\cdot\,\|$ (in particular it can be a pre-Hilbert space.) $X$ is complete if and only if for every $\{x_n\}\subset X$
$$
\sum_{n=1}^\infty\|x_n\|<\infty\implies \sum_{n=1}^\infty x_n\text{ converges.}
$$
