  We introduce a functional domain of attraction approach for stochastic
processes, which is more general than the usual one based on weak convergence.
  The distribution function G of a continuous max-stable process on [0,1] is
introduced and it is shown that G can be represented via a norm on functional
space, called D-norm. This is in complete accordance with the multivariate case
and leads to the definition of functional generalized Pareto distributions
(GPD) W. These satisfy W=1+log(G) in their upper tails, again in complete
accordance with the uni- or multivariate case.
  Applying this framework to copula processes we derive characterizations of
the domain of attraction condition for copula processes in terms of tail
equivalence with a functional GPD.
  \delta-neighborhoods of a functional GPD are introduced and it is shown that
these are characterized by a polynomial rate of convergence of functional
extremes, which is well-known in the multivariate case.
