Locally finite operator

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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.

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.

Examples[]

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

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