Locally finite operator

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

In other words, there exists a family [math]\displaystyle{ \{ V_i\vert i\in I\} }[/math] of linear subspaces of [math]\displaystyle{ V }[/math], such that we have the following:

• [math]\displaystyle{ \bigcup_{i\in I} V_i=V }[/math]
• [math]\displaystyle{ (\forall i\in I) f[V_i]\subseteq V_i }[/math]
• Each [math]\displaystyle{ V_i }[/math] 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 [math]\displaystyle{ V }[/math] whose elements are all eigenvectors of [math]\displaystyle{ f }[/math]) linear operator is locally finite, because it is the union of subspaces spanned by finitely many eigenvectors of [math]\displaystyle{ f }[/math].