Contents

Contents

Statement

Throughout, let $\mathcal{H}$ be a separable infinite-dimensional complex Hilbert space.

Theorem

(Kuiper’s theorem)
The topological unitary group $\mathrm{U}(\mathcal{H})$ in either the

or the equivalent

is contractible in that there is a left homotopy between the identity $id \;\colon\; U(\mathcal{H}) \to U(\mathcal{H})$ and the constant function $const_{\mathrm{e}} \;\colon\; U(\mathcal{H}) \to U(\mathcal{H})$.

Similarly, since the general linear group $GL(\mathcal{H})$ of bounded operators is homotopy-equivalent to $\mathrm{U}(\mathcal{H})$ by the Gram-Schmidt process, it, too, is contractible.

See the commented list of references below.

References

Proof for the norm topology on U(ℋ):

• Nicolaas Kuiper, Contractibility of the unitary group in Hilbert space, Topology, 3, 19-30 (1964)

• Luc Illusie, Contractibilité du groupe linéaire des espaces de Hilbert de dimension infinie, Séminaire Bourbaki: années 1964/65 1965/66, exposés 277-312, Séminaire Bourbaki, no. 9 (1966), Exposé no. 284, 9 p. (numdam:SB_1964-1966__9__105_0)

Proof for the strong operator topology on U(ℋ):

Direct proof for the compact-open topology on U(ℋ):

Proof that the compact-open topology agrees with the strong operator topology on U(ℋ), by which contractibility of the former follows from that of the latter: