Locally finite operator

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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:

  • \bigcup_{i\in I} V_i=V
  • (\forall i\in I) f[V_i]\subseteq V_i
  • Each V_i is finite-dimensional.

Examples[edit]

  • 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.