# Isomorphism iff Inverse Equals Adjoint

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## Theorem

Let $H, K$ be Hilbert spaces.

Let $U \in B \struct {H, K}$ be a bounded linear transformation.

Then the following are equivalent:

- $(1): \quad U$ is an isomorphism
- $(2): \quad U$ is invertible and $U^{-1} = U^*$, where $U^*$ denotes the adjoint of $U$.

## Proof

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $II.2.5$