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:

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 $f$ .

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