  In the founding paper on unbounded $KK$-theory it was established by Baaj and
Julg that the bounded transform, which associates a class in $KK$-theory to any
unbounded Kasparov module, is a surjective homomorphism (under a separability
assumption). In this paper, we provide an equivalence relation on unbounded
Kasparov modules and we thereby describe the kernel of the bounded transform.
This allows us to introduce a notion of topological unbounded $KK$-theory,
which becomes isomorphic to $KK$-theory via the bounded transform. The
equivalence relation is formulated entirely at the level of unbounded Kasparov
modules and consists of homotopies together with an extra degeneracy condition.
Our degenerate unbounded Kasparov modules are called spectrally decomposable
since they admit a decomposition into a part with positive spectrum and a part
with negative spectrum.
