Surjection that Preserves Inner Product is Linear

Theorem
Let $H, K$ be Hilbert spaces, and denote by ${\innerprod \cdot \cdot}_H$ and ${\innerprod \cdot \cdot}_K$ their respective inner products.

Let $U: H \to K$ be a surjection such that:


 * $\forall g, h \in H: {\innerprod g h}_H = {\innerprod {Ug} {Uh} }_K$

Then $U$ is a linear map, and hence an isomorphism.

Proof
Let $x, y \in H$.

Let $\alpha \in \GF$.

By surjectivity of $U$, choose $z \in H$ such that $Uz = \map U {\alpha x + y} - \paren { \alpha Ux + Uy }$.

Then:

By positivity, $Uz = {\bf 0}_K$.

Hence:
 * $\map U {\alpha x + y} = \alpha U x + U y$

Thus, $U$ is linear.