Compact Idempotent is of Finite Rank

Theorem

Let $H$ be a Hilbert space.

Let $T \in \map {B_0} H$ be a compact linear operator, and let $T$ be idempotent.

Then:

$T \in \map {B_{00} } H$

that is, $T$ is a bounded finite rank operator.

Proof

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Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text {II}.4$: Exercise $4$