Dear Colleagues,
Could you please help me in solving the problem in the pdf attachment.
Because of LATEX technical problems I attached a file containing the problem.
Best Regards.
If $\displaystyle y = \textstyle\sum\beta_je_j$ then $\displaystyle \textstyle\|x-y\|^2 = \bigl\langle x-\sum\beta_je_j, x-\textstyle\sum\beta_je_j\bigr\rangle$, so that
$\displaystyle \textstyle\|x-y\|^2 = \|x\|^2 - 2\text{Re}\sum\bigl(\langle x,e_j\rangle\beta_je_j\bigr)\overline{\beta_j} + \sum|\beta_je_j|^2.$
Also, $\displaystyle \textstyle\sum\bigl|\langle x,e_j\rangle - \beta_j\bigr|^2 = \sum|\langle x,e_j\rangle|^2 - 2\text{Re}\sum\bigl(\langle x,e_j\rangle\beta_je_j\bigr)\overline{\beta_j} + \sum|\beta_je_j|^2.$
Subtract that equation from the previous one to see that $\displaystyle \|x-y\|^2$ is smallest when $\displaystyle \textstyle\sum\bigl|\langle x,e_j\rangle - \beta_j\bigr|^2 = 0.$