  Covariance operators of random functions are crucial tools to study the way
random elements concentrate over their support. The principal component
analysis of a random function X is well-known from a theoretical viewpoint and
extensively used in practical situations. In this work we focus on local
covariance operators. They provide some pieces of information about the
distribution of X around a fixed point of the space x&#8320;. A description of
the asymptotic behaviour of the theoretical and empirical counterparts is
carried out. Asymptotic developments are given under assumptions on the
location of x&#8320; and on the distributions of projections of the data on the
eigenspaces of the (non-local) covariance operator.
