Kuiper's theorem
From Wikipedia, the free encyclopedia
In mathematics, Kuiper's theorem is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the topological space X of all linear operators 
from H to itself, which are bounded operators and invertible, is such that for any finite complex Y, there is just one homotopy class of mappings from Y to X. Here the topology on X is the norm topology of operators, and the single class must be that of constant mappings.
A corollary, also called Kuiper's theorem, is that X is contractible.
This result has important uses in topological K-theory. In finite dimensions, X would be a complex general linear group and not at all contractible because it has the topology of its maximal compact subgroup, the unitary group. There is therefore a sense in which passing to infinitely many dimensions causes much of the topology to "go away".
The result was proved by the Dutch mathematician Nicolaas Kuiper.
[edit] References
- Kuiper, N. (1965). "The homotopy type of the unitary group of Hilbert space". Topology 3: 19–30. doi:.
 |
This topology-related article is a stub. You can help Wikipedia by expanding it. |
