Linear Transformation has Finite Index iff Pseudoinverse exists

Theorem

Let $U, V$ be vector spaces over a field $K$.

Let $T: U \to V$ be a linear transformation.

Then $T$ has finite index if and only if $T$ has a pseudoinverse.

Proof

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Sources

• 2002: Peter D. Lax: Functional Analysis: $2.2$: Index of a Linear Map