, 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.
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
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
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