שינויים

$$u=\alpha_1w_1+\cdots+\alpha_kw_k$. $אם כן, $$\left \langle v,u \right \rangle=\overline{\alpha}_1\left \langle v,w_1 \right \rangle+\cdots+\overline{\alpha}_k\left \langle v,w_k \right \rangle$$

 $$\left \langle\pi_B\left(v \right ),u \right \rangle=\frac{\left \langle v,w_1 \right \rangle}{\left \| w_1 \right \|^2}\overline{\alpha}_1\left \| w_1 \right \|^2+\cdots+\frac{\left \langle v,w_k \right \rangle}{\left \| w_k \right \|^2}\overline{\alpha}_k\left \| w_k \right \|^2=$$$$=\overline{\alpha}_1\left \langle v,w_1 \right \rangle+\cdots+\overline{\alpha}_k\left \langle v,w_k \right \rangle$$