The infinite general linear group or stable general linear group is the direct limit of the inclusions GL(n,F) → GL(n+1,F) as the upper left block matrix. It is denoted by either GL(F) or GL(∞,F), and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places.
It is used in algebraic K-theory to define K1, 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 — see Kuiper's theorem.
Read more about this topic: General Linear Group
Famous quotes containing the words infinite, general and/or group:
“In natures infinite book of secrecy
A little I can read.”
—William Shakespeare (15641616)
“A writer who writes, I am alone ... can be considered rather comical. It is comical for a man to recognize his solitude by addressing a reader and by using methods that prevent the individual from being alone. The word alone is just as general as the word bread. To pronounce it is to summon to oneself the presence of everything the word excludes.”
—Maurice Blanchot (b. 1907)
“The government of the United States at present is a foster-child of the special interests. It is not allowed to have a voice of its own. It is told at every move, Dont do that, You will interfere with our prosperity. And when we ask: where is our prosperity lodged? a certain group of gentlemen say, With us.”
—Woodrow Wilson (18561924)