If I may suggest a couple edits.

1) For the high-dimensional space, we need not just a vector space, but a vector space with a scalar product. So for both the infinite dimensional polynomials, as well as for the space of functions, could you clarify how you define the scalar product? Or, equivalently, maybe it’s easier to define the norm (as long as the norm is induced by some scalar product).

2) When you say that for computational reasons, we can drop all but the first five of the coordinates in the infinite polynomial space, it is not very clear what’s going on. If interpreted naively, one would wonder what was the point of starting with infinite dimensions in the first place, if we drop all but a handful of the dimensions?

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