Is there application of the set $\prod_{i\in\mathbb R}\mathbb R$?

Question

The set $$\prod_{i\in\mathbb R}\mathbb R$$ has cardinality larger than $\mathbb R$, in most occasion, it has been a "theoretic object" with little practical use. I wonder is there any example or situation where it is treated as "the background for something to happen"? (perhaps, when do we treat it as a vector space, topological space or banach space where interesting things happen on it, or consider differentiation or integration of real functions defined on it?)