Hilbert Grassmannians as classifying spaces

Abstract

In this short work we prove that the Hilbert Grassmannians endowed with the weak topology are models for the classifying spaces of the unitary groups. As application of this result one can use Hilbert Grassmannians for the presentation of the $K$-theory of topological spaces by computing equivalences classes of homotopy equivalent maps.

19 pages. Keywords: Hilbert Grassmannians, weak topology, classifying space, $K$-theory