In other words, there exists a family of linear subspaces of , such that we have the following:
Each is finite-dimensional.
An equivalent condition only requires to be the spanned by finite-dimensional -invariant subspaces.[3][4] If is also a Hilbert space, sometimes an operator is called locally finite when the sum of the is only dense in .[2]: 78–79
^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. JSTOR2374012.
^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). CiteSeerX10.1.1.867.5629.