  Let $V$ be a complete discrete valuation ring with residue field $k$ and with
fraction field $K$ of characteristic 0. We clarify the analysis behind the
Monsky--Washnitzer completion of a commutative $V$-algebra using spectral
radius estimates for bounded subsets in complete bornological $V$-algebras.
This leads us to a functorial chain complex for commutative $k$-algebras that
computes Berthelot's rigid cohomology. This chain complex is related to the
periodic cyclic homology of certain complete bornological $V$-algebras.
