Locally finite operator - Alchetron, the free social encyclopedia

In mathematics, a linear operator f : V → 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 | i ∈ I } of linear subspaces of V , such that we have the following:

⋃ i ∈ I V i = V

( ∀ i ∈ I ) f [ V i ] ⊆ V i

Each 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 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 .

References

Locally finite operator Wikipedia

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