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 talking with scholars, I observe that they lost on ruder companions those years of boyhood which alone could give imaginative literature a religious and infinite quality in their esteem.”
—Ralph Waldo Emerson (1803–1882)
“However energetically society in general may strive to make all the citizens equal and alike, the personal pride of each individual will always make him try to escape from the common level, and he will form some inequality somewhere to his own profit.”
—Alexis de Tocqueville (1805–1859)
“There is nothing in the world that I loathe more than group activity, that communal bath where the hairy and slippery mix in a multiplication of mediocrity.”
—Vladimir Nabokov (1899–1977)