  For any $n$-tuple $(\alpha_1,...,\alpha_n)$ of linearly independent vectors
in Hilbert space $H$, we construct a unique orthonormal basis
$(\epsilon_1,...,\epsilon_n)$ of $span\{\alpha_1,...,\alpha_n\}$ satisfying:
$$\sum_{i=1}^n\|\epsilon_i-\alpha_i\|^2\le\sum_{i=1}^n\|\beta_i-\alpha_i\|^2$$
for all orthonormal basis $(\beta_1,...,\beta_n)$ of
$span\{\alpha_1,...,\alpha_n\}$.
  We study the stability of the orthornormalization and give some applications
and examples.
