# Definition:Finite Rank Operator

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## Definition

Let $H, K$ be Hilbert spaces.

Let $T: H \to K$ be a linear transformation.

Then $T$ is said to be a **finite rank operator**, or **of finite rank**, if and only if its range, $\Rng T$, is finite dimensional.

Note that a finite rank operator is not necessarily bounded.

## Linguistic Note

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As linear operator usually has a more specified meaning, the name **finite rank operator** is a case of pars pro toto.

This might be due to **finite rank linear transformation**, which is formally more correct, being awkward in pronunciation.

## Also see

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) ... (previous) ... (next) $II.4.3$