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In mathematics, a linear operator ${\displaystyle f:V\to V}$ is called locally finite if the space ${\displaystyle V}$ is the union of a family of finite-dimensional ${\displaystyle f}$-invariant subspaces.

In other words, there exists a family ${\displaystyle \{V_{i}\vert i\in I\}}$ of linear subspaces of ${\displaystyle V}$, such that we have the following:

• ${\displaystyle \bigcup _{i\in I}V_{i}=V}$
• ${\displaystyle (\forall i\in I)f\subseteq V_{i}}$
• Each ${\displaystyle V_{i}}$ is finite-dimensional.

An equivalent condition only requires ${\displaystyle V}$ to be the spanned by finite-dimensional ${\displaystyle f}$-invariant subspaces. If ${\displaystyle V}$ is also a Hilbert space, sometimes an operator is called locally finite when the sum of the ${\displaystyle \{V_{i}\vert i\in I\}}$ is only dense in ${\displaystyle V}$.

Examples

• Every linear operator on a finite-dimensional space is trivially locally finite.
• Every diagonalizable (i.e. there exists a basis of ${\displaystyle V}$ whose elements are all eigenvectors of ${\displaystyle f}$) linear operator is locally finite, because it is the union of subspaces spanned by finitely many eigenvectors of ${\displaystyle f}$.
• The operator on ${\displaystyle \mathbb {C} }$, the space of polynomials with complex coefficients, defined by ${\displaystyle T(f(x))=xf(x)}$, is not locally finite; any ${\displaystyle T}$-invariant subspace is of the form ${\displaystyle \mathbb {C} f_{0}(x)}$ for some ${\displaystyle f_{0}(x)\in \mathbb {C} }$, and so has infinite dimension.
• The operator on ${\displaystyle \mathbb {C} }$ defined by ${\displaystyle T(f(x))={\frac {f(x)-f(0)}{x}}}$ is locally finite; for any ${\displaystyle n}$, the polynomials of degree at most ${\displaystyle n}$ form a ${\displaystyle T}$-invariant subspace.

References

1. Yucai Su; Xiaoping Xu (2000). "Central Simple Poisson Algebras". arXiv:math/0011086v1.
2. ^ DeWilde, Patrick; van der Veen, Alle-Jan (1998). Time-Varying Systems and Computations. Dordrecht: Springer Science+Business Media, B.V. doi:10.1007/978-1-4757-2817-0. ISBN 978-1-4757-2817-0.
3. Radford, David E. (Feb 1977). "Operators on Hopf Algebras". American Journal of Mathematics. 99 (1). Johns Hopkins University Press: 139–158. doi:10.2307/2374012. JSTOR 2374012.
4. Scherpen, Jacquelien; Verhaegen, Michel (September 1995). On the Riccati Equations of the H Control Problem for Discrete Time-Varying Systems. 3rd European Control Conference (Rome, Italy). CiteSeerX 10.1.1.867.5629.
5. Joppy (Apr 28, 2018), answer to "Locally Finite Operator". Mathematics StackExchange. StackOverflow.

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