Kuiper's theorem

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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 $\mathsf{L}$ 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.