# Surjection that Preserves Inner Product is Linear

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## Theorem

Let $H, K$ be Hilbert spaces, and denote by $\left\langle{\cdot, \cdot}\right\rangle_H$ and $\left\langle{\cdot, \cdot}\right\rangle_K$ their respective inner products.

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

- $\forall g,h \in H: \left\langle{g, h}\right\rangle_H = \left\langle{Ug, Uh}\right\rangle_K$

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

## Proof

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next): $I.5$: Exercise $9$