  Motivated by the increasing availability of data of functional nature, we
develop a general probabilistic and statistical framework for extremes of
regularly varying random elements $X$ in $L^2[0,1]$. We place ourselves in a
Peaks-Over-Threshold framework where a functional extreme is defined as an
observation $X$ whose $L^2$-norm $\|X\|$ is comparatively large. Our goal is to
propose a dimension reduction framework resulting into finite dimensional
projections for such extreme observations. Our contribution is double. First,
we investigate the notion of Regular Variation for random quantities valued in
a general separable Hilbert space, for which we propose a novel concrete
characterization involving solely stochastic convergence of real-valued random
variables. Second, we propose a notion of functional Principal Component
Analysis (PCA) accounting for the principal `directions' of functional
extremes. We investigate the statistical properties of the empirical covariance
operator of the angular component of extreme functions, by upper-bounding the
Hilbert-Schmidt norm of the estimation error for finite sample sizes. Numerical
experiments with simulated and real data illustrate this work.
