Editing General linear group (section)

== Infinite general linear group ==
The '''infinite general linear group''' or '''[[direct limit of groups|stable]] general linear group''' is the [[direct limit]] of the inclusions <math>\operatorname{GL}(n,F)\to \operatorname{GL}(n+1,F)</math> as the upper left [[block matrix]]. It is denoted by either <math>\operatorname{GL}(F)</math> or <math>\operatorname{GL}(\infty,F)</math>, and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places.<ref name=Mil25>{{cite book | last1=Milnor | first1=John Willard | author1-link= John Milnor | title=Introduction to algebraic K-theory | publisher=[[Princeton University Press]] | location=Princeton, NJ | mr=0349811 | year=1971 | zbl=0237.18005 | series=Annals of Mathematics Studies | volume=72 | page=25 }}</ref>

It is used in [[algebraic K-theory]] to define [[Algebraic K-theory#K1|K<sub>1</sub>]], and over the reals has a well-understood topology, thanks to [[Bott periodicity]].

It should not be confused with the space of (bounded) invertible operators on a [[Hilbert space]], which is a larger group, and topologically much simpler, namely contractible &ndash; see [[Kuiper's theorem]].