  We present an approach to defining Hilbert spaces of functions depending on
infinitely many variables or parameters, with emphasis on a weighted tensor
product construction based on stable space splittings, The construction has
been used in an exemplary way for guiding dimension- and scale-adaptive
algorithms in application areas such as statistical learning theory, reduced
order modeling, and information-based complexity. We prove results on compact
embeddings, norm equivalences, and the estimation of $epsilon$-dimensions. A
new condition for the equivalence of weighted ANOVA and anchored norms is also
given.
