In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the space GL(H) of invertible bounded endomorphisms of H is such that all maps from any finite complex Y to GL(H) are homotopic to a constant, for the norm topology on operators.
In fact it is homotopy equivalent to its maximal compact subgroup, the unitary group U of H. The proof that the complex general linear group and unitary group have the same homotopy type is by the Gram-Schmidt process, or through the matrix polar decomposition, and carries over to the infinite-dimensional case of separable Hilbert space, basically because the space of upper triangular matrices is contractible as can be seen quite explicitly.
The stabiliser of a single vector v of the unit sphere is the unitary group of the orthogonal complement of v; therefore the homotopy long exact sequence predicts that all the homotopy groups of the unit sphere will be trivial.
This shows the close topological relationship, but is not in itself quite enough, since the inclusion of a point will be a weak homotopy equivalence only, and that implies contractibility directly only for a CW complex.
In a paper published two years after Kuiper's,[9] There is another infinite-dimensional unitary group, of major significance in homotopy theory, that to which the Bott periodicity theorem applies.
The difference from Kuiper's group can be explained: Bott's group is the subgroup in which a given operator acts non-trivially only on a subspace spanned by the first N of a fixed orthonormal basis {ei}, for some N, being the identity on the remaining basis vectors.