# Range of Idempotent is Kernel of Complementary Idempotent

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## Theorem

Let $H$ be a Hilbert space.

Let $A$ be an idempotent operator.

Then $\Rng A = \map \ker {I - A}$.

### Corollary 1

Furthermore:

- $\ker A = \Rng {I - A}$

### Corollary 2

$\Rng A$ is a closed linear subspace of $H$.

## Proof

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## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) ... (previous) ... (next) $\text {II}.3.2 \ \text {(b)}$