Vector space/K/Inner product/Orthonormal basis/Coefficients/Fact

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product, and let ${}u_{i}$, ${}i\in I$, be an orthonormal basis of ${}V$.

Then the coefficients of a vector ${}v$, with respect to this basis, are given by

${}v=\sum _{i\in I}\left\langle v,u_{i}\right\rangle u_{i}\,.$

Proof, Write another proof