Hilbert space-valued random elements admit Karhunen-Loève expansions precisely when they lie in the appropriate Bochner space and their covariance operator is Hilbert-Schmidt, via a natural isomorphism with computational advantages.
Numerical methods for the dis- cretization of random fields by means of the Karhunen–Lo` eve expansion
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Karhunen Lo\`eve Expansions of Hilbert Space-Valued Random Elements
Hilbert space-valued random elements admit Karhunen-Loève expansions precisely when they lie in the appropriate Bochner space and their covariance operator is Hilbert-Schmidt, via a natural isomorphism with computational advantages.