Let $(V,\langle\,,\rangle)$ be a vector space with the dot product $\langle\cdot,\cdot\rangle$. Let $f:V\to V$ be a linear map with the property $\lVert f(x)\rVert =\lVert x \rVert$ for all $x\in V$. Show that this already implies $\langle f(x),f(y) \rangle = \langle x,y \rangle$ for all $x,y \in V$
My first idea was to show this with the Cauchy–Schwarz inequality. But it didn't yield anything useful since all i get is $\lVert \langle f(x),f(y) \rangle \rVert \le ||f(x)||^2 ||f(y)||^2 =||x||^2||y||^2 = \langle x,x \rangle \langle y,y \rangle$
I would appreciate a little hint.