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To prove the schwarz inequality they proved,they showed that the vector z is orthogonal to the vector v. If they are orthogonal, then why didn't they have written in this way : $\displaystyle \|u\|^{2}=|v\|^{2}+\|z\|^{2}$?

Instead of this: $${\displaystyle \|u\|^{2}=\left|{\frac {\langle u,v\rangle }{\langle v,v\rangle }}\right|^{2}\|v\|^{2}+\|z\|^{2}}$$

Another thing is that, how they got the below? I mean, I don't see the another $\|v\|^{2}$ in the below line. $$\left|{\frac {\langle u,v\rangle }{\langle v,v\rangle }}\right|^{2}\|v\|^{2}+\|z\|^{2}={\frac {|\langle u,v\rangle |^{2}}{\|v\|^{2}}}+\|z\|^{2}$$

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First, since $v$ and $z$ are orthogonal, then $\frac {\langle u,v\rangle }{\langle v,v\rangle }v$ and $z$ are too. So, by Pythagorean theorem, $${\displaystyle \|u\|^{2}=\left|{\frac {\langle u,v\rangle }{\langle v,v\rangle }}\right|^{2}\|v\|^{2}+\|z\|^{2}}$$ The second part follows of $\|v\|^2=\langle v,v\rangle$.

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