Covariance operator
In probability theory, for a probability measure P on a Hilbert space H with inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } , the covariance of P is the bilinear form Cov: H × H → R given by C o v ( x , y ) = ∫ H ⟨ x , z ⟩ ⟨ y , z ⟩ d P ( z ) {\displaystyle \mathrm {Cov} (x,y)=\int _{H}\langle x,z\rangle \langle y,z\rangle \,\mathrm {d} \mathbf {P} (z)} for all x and y in H. The covariance operator C is then defined by C o v ( x , y ) = ⟨ C x , y ⟩ {\displaystyle \mathrm {Cov} (x,y)=\langle Cx,y\rangle } (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.