In page 121 of the original LS Paper they use the fact that the space of functions they are dealing with (payoffs of American options), belong to the $\mathcal L^2$ space.
They use this assumption to allow the following: (1) a unique orthogonal projection of this space of Americans Payoffs exist and (2) the orthogonal projection can be decomposed as a finite combination of bases.
So in the end, they come with a polynomial representation of the conditional expectation.
I have some questions regarding the $\mathcal L^2$ assumption:
1) Do we need to deal with infinite dimensional spaces?
1.1.) Our Monte-Carlo simulation gives us a finite number of vectors, therefore aren't we on finite spaces?
1.2) If we are on finite spaces, I want to confirm that in the finite space case, we will always find an orthogonal projection and we will always find a decomposition on countable number of bases.
2) In the case that we want to insist with infinite-dimension spaces:
2.1) we know that Americans are convex functions, but how do we prove that they are they of bounded variance (square integrable)? I read the papers that they mention Karatzas and Bensoussan but it's still not clear to me how to prove that Americans payoff can form an $\mathcal L^2$ space
2.3) Then the authors continue to work on a portfolio of many path dependent options (as their example of American-Bermudan-Asian)? how do we prove they are of bounded variation?
2.2) Now, a question regarding the subspace used in the regression (their "Xs" i.e. the stock prices).
In the case of orthogonal projections, we not only need to show that we are starting with a Hilbert space, but also, that the space we are projecting onto, is a subspace of it (i.e. it's a closed subspace of the "Ys" - the american prices). My understanding is that the subspace is the observed stock prices, how do we know they form a subspace of the original Hilbert space?