Least Squares Problems in Orthornormalization

Abstract: 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.