The Cartan-Dieudonne in $\mathbb{R}^n$ states that every orthogonal $n\times n$ matrix can be written as a product of at most $n$ reflections along hyperplanes.
My question is this: Is there a counterpart for infinite-dimensional real Hilbert space? Note that reflections along hyperplanes are defined there. Which isometries are (pointwise) limits of products of reflections? I assume this has been studied but I couldn't find anything. Any comments/reference would be appreciated!
The Cartan-Dieudonne theorem fails in infinite dimensions. An explanation given in Advances in Ultrametric Analysis: 12th International Conference on $p$-adic Functional Analysis, page 255, is that in infinite dimensions the isometry $-{\rm Id}$ cannot be written as a finite product of hyperplane reflections.