Let $H$ be an infinite dimensional Hilbert space with inner product $\langle \cdot, \cdot\rangle$ and norm $\|\cdot\|_H$. This norm induces a distance on $H$ given by $$ \forall p,q\in H \ d_H(q,p)=\|q-p\|_H $$
Let $\Omega \subset \mathbb{R}^n$ be compact and $f$ be a map $$ f: \Omega \longrightarrow H $$ such that it is continuous with respect to the euclidean standard metric $d_E$ in $\mathbb{R}^n$ and the metric $d_H$ in $H$. Moreover, the partial derivatives of $f$ with respect to each component of the coordinates in $\Omega$ exists and $$ \frac{\partial f}{\partial x_i} \in H $$
Under which circumstances $f$ defines a finite dimensional manifold in $H$ with respect to the topology induced by the metric $d_H$?
If $f$ is neither an injection or a surjection, can the image of $\Omega$ under $f$ be endowed with a manifold structure?
Is there a more convenient topology on $H$ that allows to define a manifold?
Motivation
Interpolating polynomials are an important tool in several Engineering applications. An important case is when the domain of such polynomials is a fixed interval $I\subset \mathbb{R}$ and their codomain is a compact set $\Omega\subset\mathbb{R}^n$. If we set a fixed sequence of $N+1$ points in the domain $\{t_i\}_{i=0}^{N}\subset I$ then any sequence of $N+1$ points on the codomain $\{w_i\}_{i=0}^{N}\subset \Omega$ defines uniquely a polynomial $p:I\longrightarrow\mathbb{R}^n$
$$ p(t) = w_0 \ell_0(t) + \ldots + w_N \ell_N(t), $$ where $\ell_i$ are the Lagrange polynomials generated by the points $\{t_i\}_{i=0}^{N}\subset I$. Because polynomials with compact domain $I$ are square integrable, the interpolation process defines a map $f:\Omega^{(N+1)}\longrightarrow L^2(I, \mathbb{R}^n)$.
If the interpolating map $f$ induces a manifold on $L^2$ then it would be possible to solve problems associated with the interpolation problem using the tools of differential geometry.
