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:
“That man’s best works should be such bungling imitations of Nature’s infinite perfection, matters not much; but that he should make himself an imitation, this is the fact which Nature moans over, and deprecates beseechingly. Be spontaneous, be truthful, be free, and thus be individuals! is the song she sings through warbling birds, and whispering pines, and roaring waves, and screeching winds.”
—Lydia M. Child (1802–1880)
“The general fact is that the most effective way of utilizing human energy is through an organized rivalry, which by specialization and social control is, at the same time, organized co-operation.”
—Charles Horton Cooley (1864–1929)
“No other group in America has so had their identity socialized out of existence as have black women.... When black people are talked about the focus tends to be on black men; and when women are talked about the focus tends to be on white women.”
—bell hooks (b. c. 1955)