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 nature’s infinite book of secrecy
A little I can read.”
—William Shakespeare (1564–1616)
“It was the words “descended into Hades”
That seemed too pagan to our liberal youth.
You know they suffered from a general onslaught.
And well, if they weren’t true why keep right on
Saying them like the heathen? We could drop them.”
—Robert Frost (1874–1963)
“He hung out of the window a long while looking up and down the street. The world’s second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
—John Dos Passos (1896–1970)