If $D\in \mathbb{C}^m$, $f,g:D\rightarrow\mathbb{C}^n$ are holomorphic maps such that \begin{equation*} \langle f(z),f(z)\rangle=\langle g(z),g(z)\rangle ,\forall z\in D\end{equation*}then there exists a unitary transformation $U$, such that $U(f(z))=g(z),\forall z\in D$.
I have proved that for any two sequences $\{e_k\}_{k=1}^{\infty},\{f_k\}_{k=1}^{\infty}\subset\mathbb{C^n}$ such that \begin{equation*} \langle e_k,e_l\rangle=\langle f_k,f_l\rangle ,\forall k,l\in \mathbb{N}\end{equation*} there exists a unitary transformation $U$, such that $U(e_k)=f_k,\forall k\in \mathbb{N}$.
My question is: does it helps to prove the problem? Or how can I prove it?