Surjective map onto a Hilbert space

Question

Let $E_0$ and $E$ be arbitrary sets, $\eta$ is a continuous and surjective map from $E_0$ to $E$.

Suppose that $\mathcal{H}$ is Hilbert space of real-valued functions on $E$, and define $\mathcal{H}^{(\eta)}$ by $\{g\circ \eta:g\in\mathcal{H}\}$ to be a space of real-valued functions on $E_0$, then

can we say that $\mathcal{H}^{(\eta)}$ is a Hilbert space of real-valued functions on $E_0$?

Answer 1

The answer is no.

Take $E_0 = \mathbb N, E = \{1\}$ and $\eta(n) = 1$ for every $n\in\mathbb N$. Then, $\mathcal H = \ell^2(E) = \mathbb R$ and $\mathcal H^\eta$ isn't even a subset of $\ell^2(E_0)$.