Covariance operator

In probability theory, for a probability measure P on a Hilbert space "H" with inner product formula_1, the covariance of P is the bilinear form Cov: "H" × "H" → R given by
for all "x" and "y" in "H". The covariance operator "C" is then defined by
(from the Riesz representation theorem, such operator exists if Cov is bounded). Since Cov is symmetric in its arguments, the covariance operator is
self-adjoint (the infinite-dimensional analogy of the transposition symmetry in the finite-dimensional case). When P is a centred Gaussian measure, "C" is also a nuclear operator. In particular, it is a compact operator of trace class, that is, it has finite trace.
Even more generally, for a probability measure P on a Banach space "B", the covariance of P is the bilinear form on the algebraic dual "B"#, defined by
where formula_5 is now the value of the linear functional "x" on the element "z".
Quite similarly, the covariance function of a function-valued random element (in special cases is called random process or random field) "z" is
where "z"("x") is now the value of the function "z" at the point "x", i.e., the value of the linear functional formula_7 evaluated at "z".