Locally finite operator

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

In other words, there exists a family ${\displaystyle \{V_{i}\vert i\in I\}}$ of linear subspaces of $V$, 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 $V_{i}$ is finite-dimensional.

Examples

• Every linear operator on a finite-dimensional space is trivially locally finite.
• Every diagonalizable (i.e. there exists a basis of $V$ whose elements are all eigenvectors of $f$) linear operator is locally finite, because it is the union of subspaces spanned by finitely many eigenvectors of $f$.