Covariance operator

In probability theory, for a probability measure P on a Hilbert space H with inner product ⟨ ⋅ , ⋅ ⟩ {displaystyle langle cdot ,cdot angle } , the covariance of P is the bilinear form Cov: H × H → R given by In probability theory, for a probability measure P on a Hilbert space H with inner product ⟨ ⋅ , ⋅ ⟩ {displaystyle langle cdot ,cdot angle } , 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 isself-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 ⟨ x , z ⟩ {displaystyle langle x,z angle } 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 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 u ↦ u ( x ) {displaystyle umapsto u(x)} evaluated at z.