Locally finite operator

In mathematics, a linear operator is called locally finite if the space is the union of a family of finite-dimensional -invariant subspaces.
In other words, there exists a family of linear subspaces of, such that we have the following:

An equivalent condition only requires to be spanned by finite-dimensional -invariant subspaces. If is also a Hilbert space, sometimes an operator is called locally finite when the sum of the is only dense in.

Examples

Every linear operator on a finite-dimensional space is trivially locally finite.
Every diagonalizable linear operator is locally finite, because it is the union of subspaces spanned by finitely many eigenvectors of.
The operator on, the space of polynomials with complex coefficients, defined by, is not locally finite; any -invariant subspace is of the form for some, and so has infinite dimension.
The operator on defined by is locally finite; for any, the polynomials of degree at most form a -invariant subspace.