The Thom isomorphism in bivariant K-theory

We give a simple proof of the smooth Thom isomorphism for complex bundles for the bivariant K-theories on locally convex algebras considered by Cuntz. We also prove the Thom isomorphism in Kasparov’s KK-theory in a form stated without proof in the conspectus. Along the way, we prove Bott periodicity directly on R^n, using for the Kasparov product the operator that also appears in recent work of Wulkenhaar on non-compact spectral triples with finite volume, and which may be seen as a unitalisation of the Dirac-element.