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In probability theory, for a probability measure **P** on a Hilbert space *H* with inner product , 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 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 evaluated at *z*.

- Abstract Wiener space – Mathematical construction relating to infinite-dimensional spaces.
- Cameron–Martin theorem – Theorem defining translation of Gaussian measures (Wiener measures) on Hilbert spaces.
- Feldman–Hájek theorem – Theory in probability theory
- Structure theorem for Gaussian measures – Mathematical theorem

- Baker, C. R. (September 1970).
*On Covariance Operators*. Mimeo Series. Vol. 712. University of North Carolina at Chapel Hill. - Baker, C. R. (December 1973). "Joint Measures and Cross-Covariance Operators" (PDF).
*Transactions of the American Mathematical Society*.**186**: 273–289. - Vakhania, N. N.; Tarieladze, V. I.; Chobanyan, S. A. (1987). "Covariance Operators".
*Probability Distributions on Banach Spaces*. Dordrecht: Springer Netherlands. pp. 144–183. doi:10.1007/978-94-009-3873-1_3. ISBN 978-94-010-8222-8. Retrieved 2024-04-11.