Vector space/K/Inner product/Finite-dimensional/Orthonormalization/Fact/Proof

Proof

We prove the statement by induction over ${}i$, that is, we construct successively a family of orthonormal vectors spanning the same linear subspaces. For ${}i=1$, we just have to normalize ${}v_{1}$, that is, we replace it by ${}u_{1}={\frac {v_{1}}{\Vert {v_{1}}\Vert }}$. Now suppose that the statement is already proven for ${}i$. Let a family of orthonormal vectors ${}u_{1},\ldots ,u_{i}$ fulfilling ${}\langle u_{1},\ldots ,u_{i}\rangle =\langle v_{1},\ldots ,v_{i}\rangle$ be already constructed. We set

$w_{i+1}=v_{i+1}-\left\langle v_{i+1},u_{1}\right\rangle u_{1}-\cdots -\left\langle v_{i+1},u_{i}\right\rangle u_{i}\,.$

Due to

${}{\begin{aligned}\left\langle w_{i+1},u_{j}\right\rangle &=\left\langle v_{i+1}-\left\langle v_{i+1},u_{1}\right\rangle u_{1}-\cdots -\left\langle v_{i+1},u_{i}\right\rangle u_{i},u_{j}\right\rangle \\&=\left\langle v_{i+1},u_{j}\right\rangle -\sum _{k\leq i,k\neq j}\left\langle v_{i+1},u_{k}\right\rangle \left\langle u_{k},u_{j}\right\rangle -\left\langle v_{i+1},u_{j}\right\rangle \left\langle u_{j},u_{j}\right\rangle \\&=\left\langle v_{i+1},u_{j}\right\rangle -\left\langle v_{i+1},u_{j}\right\rangle \\&=0,\end{aligned}}$

this vector is orthgonal to all ${}u_{1},\ldots ,u_{i}$, and also

${}\langle u_{1},\ldots ,u_{i},w_{i+1}\rangle =\langle v_{1},\ldots ,v_{i},v_{i+1}\rangle \,$

holds. By normalizing ${}w_{i+1}$, we obtain ${}u_{i+1}$.

To fact