Locally finite operator

In mathematics, a linear operator ${\displaystyle f:V\to V}$ is called locally finite if the space ${\displaystyle V}$ is the union of a family of finite-dimensional ${\displaystyle f}$-invariant subspaces.^: 40

In other words, there exists a family ${\displaystyle \{V_{i}\vert i\in I\}}$ of linear subspaces of ${\displaystyle V}$, such that we have the following:

• ${\displaystyle \bigcup _{i\in I}V_{i}=V}$
• ${\displaystyle (\forall i\in I)f[V_{i}]\subseteq V_{i}}$
• Each ${\displaystyle V_{i}}$ is finite-dimensional.

An equivalent condition only requires ${\displaystyle V}$ to be the spanned by finite-dimensional ${\displaystyle f}$-invariant subspaces. If ${\displaystyle V}$ is also a Hilbert space, sometimes an operator is called locally finite when the sum of the ${\displaystyle \{V_{i}\vert i\in I\}}$ is only dense in ${\displaystyle V}$.^: 78–79