for some $k$ vectors in $V, V=\operatorname{Sp}\lbrace u_i\rbrace $ if $\forall v\in V, \|v\|^{2}=\sum_{i=1}^{k}|\langle v,u_i\rangle|^2$

Question

Let $\lbrace u_1,\dots,u_k\rbrace\in V$ be some vectors in an inner product space, $\dim V=n\geq k$. if $\forall v\in V, \|v\|^2=\sum_{i=1}^k|\langle v,u_i\rangle |^2$, prove that they span $V$.

I tried various things: CS inequality, writing the inner products as double-sums (a mess) of coordinates of $v$, proving by negation (if there are $\ell>k$ vectors needed to span $V$). Couldn't sort it out. What am I missing?

Answer 1

Suppose that $U=\{u_1,...,u_k\}$ does not span $V$ and let $u$ be a non zero element of $V$ orthogonal to $U$, $\|u\|^2\neq 0$, but $\sum_{i=1}^{i=k} |\langle u,u_i\rangle|=0$.