Locally finite operator

In mathematics, a linear operator $f: V\to V$ 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 $\{ V_i\vert i\in I\}$ of linear subspaces of $V$, such that we have the following:

• $\bigcup_{i\in I} V_i=V$
• $(\forall i\in I) f[V_i]\subseteq V_i$
• 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 $f$.