Range of Idempotent is Kernel of Complementary Idempotent/Corollary 2

Corollary to Range of Idempotent is Kernel of Complementary Idempotent

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.

Let $A$ be an idempotent operator.

Then:

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

Proof

From Range of Idempotent is Kernel of Complementary Idempotent, we have:

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

From Space of Bounded Linear Transformations forms Vector Space:

$I - A$ is a Definition:Bounded Linear Transformation.

From Kernel of Bounded Linear Transformation is Closed Linear Subspace, we have:

$\map \ker {I - A}$ is a closed linear subspace of $\HH$.

Hence $\Rng A$ is a closed linear subspace of $\HH$.

$\blacksquare$

Sources

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