"$H=W$" in infinite dimensions

TL;DR

This paper extends the classical $H=W$ theorem to infinite-dimensional spaces, specifically $ ext{l}^2$, by proving the density of smooth functions in Sobolev spaces, which simplifies analysis of differential operators.

Contribution

It establishes an infinite-dimensional $H=W$ theorem, proving smooth functions are dense in Sobolev spaces on certain open sets in $ ext{l}^2$, under the segment condition.

Findings

Smooth functions are dense in Sobolev spaces on open sets in $ ext{l}^2$ satisfying the segment condition.

Density results facilitate deriving $L^2$ estimates for differential operators in infinite dimensions.

Approximation by smooth cylindrical functions holds on open sets with the segment condition, enabling calculus-based analysis.

Abstract

The classical ``$H=W$" theorem establishes the identity between two function spaces on an arbitrary nonempty open set in the Euclidean spaces: the space $W$ defined via weak derivatives, and the space $H$ defined as the closure of smooth functions within $W$ space. Extending this result to infinite-dimensional spaces is challenging due to the lack of a nontrivial translation-invariant measure and the proliferation of infinite sums inherent to infinite dimensions. In this paper, by adapting several techniques developed in our previous works, we prove that smooth functions are dense in the Sobolev space of functions on arbitrary non-empty open set in $\ell^2$, thereby establishing an infinite-dimensional counterpart of ``$H=W$". Such density results reduce the problem of deriving a priori $L^2$ estimates for differential operators -- originating from the classical Fredholm alternative and Carleman estimates -- to the simpler case of smooth functions. If approximation by smooth cylindrical functions is possible, the problem can be reduced to calculus. Unfortunately, this does not hold for every open set in $\ell^2$. However, we prove that such an approximation does hold on open sets that satisfy the segment condition.