Compact Idempotent is of Finite Rank
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This theorem requires a proof..
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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 ... (previous) ... (next): $\text {II}.4$: Exercise $4$