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

Gram-Schmidt orthonormalization

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with an inner product, and let ${}v_{1},v_{2},\ldots ,v_{n}$ be a basis of ${}V$.

Then there exists an

orthonormal basis ${}u_{1},u_{2},\ldots ,u_{n}$ of ${}V$ with

${}\langle v_{1},v_{2},\ldots ,v_{i}\rangle =\langle u_{1},u_{2},\ldots ,u_{i}\rangle \,$

for all

${}i=1,\ldots ,n$.

Proof, Write another proof