This is an announcement for the paper "Least squares problems in orthornormalization" by Shanwen Hu. Abstract: For any $n$-tuple $(\alpha_1,\cdots,\alpha_n)$ of linearly independent vectors in Hilbert space $H$, we construct a unique orthonormal basis $(\epsilon_1,\cdots,\epsilon_n)$ of $span\{\alpha_1,\cdots,\alpha_n\}$ satisfying: $$\sum_{i=1}^n\|\epsilon_i-\alpha_i\|^2\le\sum_{i=1}^n\|\beta_i-\alpha_i\|^2$$ for all orthonormal basis $(\beta_1,\cdots,\beta_n)$ of $span\{\alpha_1,\cdots,\alpha_n\}$. We study the stability of the orthornormalization and give some applications and examples. Archive classification: math.FA Remarks: 10 pages Submitted from: swhu@math.ecnu.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1210.7400 or http://arXiv.org/abs/1210.7400