Inner product/K/Cauchy-Schwarz/Fact/Proof

Proof

For ${}w=0$ , the statement holds. So suppose ${}w\neq 0$ , hence, also ${}\Vert {w}\Vert \neq 0$ holds. Therefore, we have the estimates

{}{\begin{aligned}0&\leq \left\langle v-{\frac {\left\langle v,w\right\rangle }{\Vert {w}\Vert ^{2}}}w,v-{\frac {\left\langle v,w\right\rangle }{\Vert {w}\Vert ^{2}}}w\right\rangle \\&=\left\langle v,v\right\rangle -{\frac {\left\langle v,w\right\rangle }{\Vert {w}\Vert ^{2}}}\left\langle w,v\right\rangle -{\frac {\overline {\left\langle v,w\right\rangle }}{\Vert {w}\Vert ^{2}}}\left\langle v,w\right\rangle +{\frac {\left\langle v,w\right\rangle {\overline {\left\langle v,w\right\rangle }}}{\Vert {w}\Vert ^{4}}}\left\langle w,w\right\rangle \\&=\left\langle v,v\right\rangle -{\frac {\left\langle v,w\right\rangle }{\Vert {w}\Vert ^{2}}}{\overline {\left\langle v,w\right\rangle }}-{\frac {\overline {\left\langle v,w\right\rangle }}{\Vert {w}\Vert ^{2}}}\left\langle v,w\right\rangle +{\frac {\left\langle v,w\right\rangle {\overline {\left\langle v,w\right\rangle }}}{\Vert {w}\Vert ^{2}}}\\&=\left\langle v,v\right\rangle -{\frac {\left\langle v,w\right\rangle {\overline {\left\langle v,w\right\rangle }}}{\Vert {w}\Vert ^{2}}}\\&=\left\langle v,v\right\rangle -{\frac {\vert {\left\langle v,w\right\rangle }\vert ^{2}}{\Vert {w}\Vert ^{2}}}.\end{aligned}}

Multiplication with ${}\Vert {w}\Vert ^{2}$ and taking the square root yields the result.