Karhunen Lo\`eve Expansions of Hilbert Space-Valued Random Elements

The Karhunen-Lo\`eve Expansion (KLE) of a stochastic process is a well understood eigenfunction expansion used widely in time series analysis, stochastic PDEs, and signal processing. Karhunen-Lo\`eve expansions have also been proven to exist for other types of stochastic elements whose values lie in certain $L^2$ spaces. This article provides a concise proof about the necessary and sufficient conditions for a function $v$ defined on some sample space $\Omega$ and whose values lie in some Hilbert space $\mathcal H$ to admit an eigenfunction expansion like the well-known KLE. We draw on the existing theory of Bochner spaces and Hilbert-Schmidt spaces and construct an isomorphism between them. Furthermore, this isomorphism is natural, which has important computational consequences. Finally, we demonstrate with an example the computational advantages conferred by considering the KLE in this generalized setting.