Least Squares Problems in Orthornormalization

For any $n$-tuple $(α_1,...,α_n)$ of linearly independent vectors in Hilbert space $H$, we construct a unique orthonormal basis $(ε_1,...,ε_n)$ of $span\{α_1,...,α_n\}$ satisfying: $\sum_{i=1}^n\|ε_i-α_i\|^2\le\sum_{i=1}^n\|β_i-α_i\|^2$ for all orthonormal basis $(β_1,...,β_n)$ of $span\{α_1,...,α_n\}$. We study the stability of the orthornormalization and give some applications and examples.

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