Definition:Linear Transformation of Finite Index

Definition

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

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

$T$ is said to have finite index if and only if:

$\paren 1$ $\map \ker T$ is finite-dimensional

$\paren 2$ the quotient space $V / \Img T$ is finite-dimensional

where:

$\map \ker T$ denotes the kernel of $T$

$\Img T$ denotes the image of $T$

Also see

• Definition:Index of Linear Transformation
• Definition:Pseudoinverse of Linear Transformation
• Linear Transformation has Finite Index iff Pseudoinverse exists

Sources

• 2002: Peter D. Lax: Functional Analysis: Chapter $27$: Index Theory