  We prove excision in entire and periodic cyclic cohomology and construct a
Chern-Connes character for Fredholm modules over a C*-algebra without
summability restrictions, taking values in a variant of Connes's entire cyclic
cohomology.
  Before these results can be obtained, we have to sort out some fundamental
questions about the class of algebras on which to define entire cyclic
cohomology. The right domain of definition for entire cyclic cohomology is the
category of complete bornological algebras. For these algebras, we define a
bivariant cohomology theory, called analytic cyclic cohomology, that contains
Connes's entire cyclic cohomology as a special case.
  The definition of analytic cyclic cohomology is based on the Cuntz-Quillen
approach to cyclic cohomology theories using tensor algebras and X-complexes.
The appropriate completion of the tensor algebra that yields analytic cyclic
cohomology can be understood using an appropriate notion of analytic
nilpotence.
  In addition, we develop the elementary theory of analytic cyclic cohomology
(smooth homotopy invariance, stability, Chern character in K-theory).
