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 of linear subspaces of $V$, $\{V_{i}\vert i\in I\}$ 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

• any linear operator on a finite-dimensional space is trivially locally finite
• any diagonalizable linear operator is locally finite, because it is the union of subspaces spanned by finitely many eigenvectors of $f$
• as a consequence of the above and the spectral theorem, any compact, self-adjoint operator on a Hilbert space is locally finite