  The literature on time series of functional data has focused on processes of
which the probabilistic law is either constant over time or constant up to its
second-order structure. Especially for long stretches of data it is desirable
to be able to weaken this assumption. This paper introduces a framework that
will enable meaningful statistical inference of functional data of which the
dynamics change over time. We put forward the concept of local stationarity in
the functional setting and establish a class of processes that have a
functional time-varying spectral representation. Subsequently, we derive
conditions that allow for fundamental results from nonstationary multivariate
time series to carry over to the function space. In particular, time-varying
functional ARMA processes are investigated and shown to be functional locally
stationary according to the proposed definition. As a side-result, we establish
a Cram\'er representation for an important class of weakly stationary
functional processes. Important in our context is the notion of a time-varying
spectral density operator of which the properties are studied and uniqueness is
derived. Finally, we provide a consistent nonparametric estimator of this
operator and show it is asymptotically Gaussian using a weaker tightness
criterion than what is usually deemed necessary.
