Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 8 / 7 / a / Solution

Fix a separable infinite-dimensional complex Hilbert space $\mathcal{H}$ and letwith its operator-norm topology. The odd topological K-theory group of a compact metric space is defined by continuous-map homotopy classesHomotopies use the same norm topology. Its finite-dimensional stable realization and group operation are established in the following parts, so this definition does not yet assume that homotopy classes form a group.