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:
“One who pressed forward incessantly and never rested from his labors, who grew fast and made infinite demands on life, would always find himself in a new country or wilderness, and surrounded by the raw material of life. He would be climbing over the prostrate stems of primitive forest-trees.”
—Henry David Thoreau (1817–1862)
“Mr. Chadband is a large yellow man, with a fat smile, and a general appearance of having a good deal of train oil in his system.”
—Charles Dickens (1812–1870)
“...Women’s Studies can amount simply to compensatory history; too often they fail to challenge the intellectual and political structures that must be challenged if women as a group are ever to come into collective, nonexclusionary freedom.”
—Adrienne Rich (b. 1929)