Conditions for when the future-distribution inner-product space is a Hilbert space

Determine the necessary and sufficient conditions under which the space of future distributions G, defined as the linear span of functions g_{\bar{a}} representing conditional future distributions for a discrete-time, stationary stochastic process with a countable observation set, becomes a Hilbert space when endowed with the inner product induced by the L^2(\mu_\epsilon) embedding via densities \gamma(g) constructed through the Radon–Nikodym theorem.

Background

The thesis focuses on observable operator models (OOMs) for discrete-time, stationary processes with countable observations. A central technical step toward an approximation theory is to equip the space of future distributions G with an inner product and ensure continuity of observable operators with respect to the induced norm.

The authors recount that an unpublished tutorial proposed an inner product on G but lacked proofs. They set two objectives: to recover the missing details and to investigate when G, with this inner product, is a Hilbert space. This question is posed explicitly as open in the introduction and is motivated by two approximation pathways that require a Hilbert space structure.