Editing Invariant subspace (section)

== Lattice of subspaces ==
As the above examples indicate, the invariant subspaces of a given linear transformation ''T'' shed light on the structure of ''T''. When ''V'' is a finite-dimensional vector space over an [[algebraically closed field]], linear transformations acting on ''V'' are characterized (up to similarity) by the [[Jordan canonical form]], which decomposes ''V'' into invariant subspaces of ''T''. Many fundamental questions regarding ''T'' can be translated to questions about invariant subspaces of ''T''.  

The set of {{Mvar|T}}-invariant subspaces of {{Mvar|V}} is sometimes called the '''invariant-subspace lattice''' of {{Mvar|T}} and written {{Math|Lat(''T'')}}.  As the name suggests, it is a ([[Modular lattice|modular]]) [[lattice (order)|lattice]], with [[Join and meet|meets and joins]] given by (respectively) [[set intersection]] and [[linear span]].  A [[minimal element]] in {{Math|Lat(''T'')}} in said to be a '''minimal invariant subspace'''.

In the study of infinite-dimensional operators, {{Math|Lat(''T'')}} is sometimes restricted to only the [[Closed (mathematics)|closed]] invariant subspaces.