Let $W^{2,2}(\Omega_i)$, $\Omega_i = [-1,1]$, $i = 1,\ldots,d$ be the Sobolev spaces of twice weakly differentiable, square integrable functions. Let further $\otimes_a$ denote the algebraic tensor product (in other words, the non-topological one) and $\Omega = \Omega_1 \times \ldots \times \Omega_d = [-1,1]^d$.

All functions in the space $\mathcal{W} := W^{2,2}(\Omega_1) \otimes_a \ldots \otimes_a W^{2,2}(\Omega_d)$ are continuous, i.e. $\mathcal{W} \subset C^0(\Omega)$. The induced norm $\|\cdot\|_{\mathcal{W}}$ is stronger than the norm $\|\cdot\|_{2,2}$ on $W^{2,2}(\Omega)$.

Is $\mathcal{W}$ dense in $W^{2,2}(\Omega)$ (with respect to the norm $\|\cdot\|_{2,2}$). Is hence $\overline{\mathcal{W}}^{\|\cdot\|_{2,2}} = W^{2,2}(\Omega)$?

Is the semi norm $|\cdot|_{2,2}$, defined via $|u|_{2,2}^2 := \sum_{|\alpha| = 2} \|D^{\alpha} u\|_{L^2}^2$, equivalent to $\|\cdot\|_{2,2}$ on the quotient space of functions that only differ by a constant function?

Let $r \in \mathbb{N}$, $$\mathcal{W}_r := \bigcap_{\mu = 1}^{d-1} \{ \phi \in \mathcal{W} \mid \exists g_i \in W^{2,2}(\Omega_1) \otimes_a \ldots \otimes_a W^{2,2}(\Omega_\mu), \exists h_i \in W^{2,2}(\Omega_{\mu+1}) \otimes_a \ldots \otimes_a W^{2,2}(\Omega_d): \phi(x) = \sum_{i = 1}^r g_i(x_1,\ldots,x_\mu) \cdot h_i(x_{\mu+1},\ldots,x_d), \forall x \in \Omega \}$$ the space of (tensor train) rank $r$ functions in $\mathcal{W}$. Then this space $\mathcal{W}_r$ is closed under $\|\cdot\|_{2,2}$. Let further $p_i \in \Omega, \ i = 1,\ldots,m$, for some $m \in \mathbb{N}$, and $y \in \mathbb{R}^m$. Is the problem $$u^\ast = \mathrm{argmin}_{u \in \mathcal{W}_r} \sum_{i = 1}^m (u(p_i) - y_i)^2 + |u|_{2,2}^2$$ hence well defined (based on the this propert of $\mathcal{W}_r$, and does there exist such a minimizer (regardless of uniqueness)?

I strongly believe that all three assertion hold true, but I am unaware of easy ways to prove such. As this is a bit out of topic for me, I also have a bit of trouble judging what counts as rigorous argument. I have seen variants of these questions, but could not find a specific enough answer.

I have found a remark about the validity of $1.$ in a tensor calculus book, but no proof is provided. Point $2.$ should be provided by the Poincaré inequality, although it needs to be applied on two levels, and the devil is a bit in the detail. The first part of item $3.$ requires knowledge about tensor formats, so it might be hard to answer, but just under the assumption that $\mathcal{W}_r$ is closed, the second part is still of great interest to me. I of course appreciate any help regarding only any one of the three questions.