For embedded submanifold, specifically with ambient space being $\mathbb{R}^{n}$, there are many nice properties and results. Specifically there are many examples of matrix manifolds such as the Stiefel manifold, or fixed rank matrices. I'm interested in how much generalizes when working with infinite matrices in $\mathbb{R}^{\mathbb{N}}$

Specifically there are natural examples where the basis are chosen to be (similar as) monomial basis of real polynomials. We have the infinite moment matrix for instance, which is easier to work with than the truncated one.

For instance, I know $$\mathcal{M}:= \left \{ \begin{bmatrix} c_{1} & c_{2} & \dots & c_{r} \\ c_{1}x_{1} & c_{2}x_{2} & \dots & c_{r}x_{r} \\ c_{1}x_{1}^{2} & c_{2}x_{2}^{2} & \dots & c_{r}x^{2} \\ \vdots & \vdots & \ddots & \vdots \\ c_{1}x_{1}^{n} & c_{2}x_{2}^{n} & \dots & c_{r}x_{r}^{n} \end{bmatrix} : \sum _{i}c_{i} = 1, c_{i} > 0, x_{i} \in \mathbb{R} \right\} $$ is an embedded submanifold of $\mathbb{R}^{n \times r}$, with dimension $2r-1$.

But what about the case when $n \to \infty$? Is the infinite matrix also a $2r-1$ dimension manifold? Theorems such as regular value theorem applies when the embedding space is a standard manifold, but what about $\mathbb{R}^{\mathbb{N}}$. Certainly when equipped with product topology, it is metrizable, but not sure how it makes it easier to work with.

Even if it is a manifold, I'm not sure other things such as retractions ($i$.$e$. a map from tangent space to the manifold that preserves first and second order derivatives for instance) is easy to work with at all.

So are there any techniques to study such finite dimensional manifolds?

I've heard of Banach manifolds, but have no experience with it.