Kuiper's Theorem is a result on the topology of operators on an infinite-dimensional complex Hilbert space, named for Nicolaas Kuiper. It states that the group of invertible bounded endomorphisms of such a space, given the norm topology, is topologically trivial in a strong sense: every continuous map into it from any finite-dimensional complex space is homotopic to a constant map, a striking contrast with the topology of the corresponding group in finite dimensions.
Facts
StatementThe group of invertible bounded endomorphisms of an infinite dimensional complex Hilbert space, given the norm topology, is such that every continuous map into it from any finite dimensional complex space is homotopic to a constant map. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Source Kuiper's theorem, Wikipedia
Proved By
Source Kuiper's theorem, Wikipedia
Sources
1. Kuiper's theorem, Wikipedia
Introduction
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.
References
Kuiper, N. (1965). The homotopy type of the unitary group of Hilbert space.
In Group: Algebraic Topology, Lead
In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H.
In Branch: Topology, Lead sentence
heorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H.
Proved By: Nicolaas Kuiper, Lead
The result on the group of bounded operators was proved by the Dutch mathematician Nicolaas Kuiper, for the case of a separable Hilbert space; the restriction of separability was later lifted.
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