Locally finite operator

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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:

  • Each is finite-dimensional.


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