In mathematics, Kuiper's theorem is a result on the topology of the operators on an infinite-dimensional complex Hilbert space H. It states that the topological space X of all linear operators 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. It is a corollary, itself often called Kuiper's theorem, that X is contractible. This result has important uses in topological K-theory. In finite dimensions X would be a complex general linear group, not at all contractible (it has the topology of its maximal compact subgroup, the unitary group). There is therefore a sense in which, by passing to infinitely many dimensions, much of the topology of operators 'goes away'. The result was proved by the Dutch mathematician Nicolaas Kuiper (1920-1994).
