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.
where is now the value of the linear functional x on the element z.
where z(x) is now the value of the function z at the point x, i.e., the value of the linear functional evaluated at z.
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