This question Contractibility of the sphere and Stiefel manifolds of a separable Hilbert space mentions the Stieffel manifolds of orthonormal $n$-frames $$V_n=\lbrace x_1,\dots,x_n\in \mathscr{H} \mid i\neq j\Rightarrow\langle x_i|x_j\rangle=0\text{ and }\|x_j\|=1,\forall j\rbrace$$ of a infinite dimensional separable Hilbert space $\mathscr{H}$.
How is $V_n$ a manifold? Is it some sort of a submersion level set theorem?
There are some books in the question Reference on Infinite Dimensional Manifold, but I haven't been able to find an explanation of specific examples (and if I find a suitable reference I expect there'd be also Grassmannian)...
