Let $A$ be a dense subspace of a Hilbert space $H$. Denote $\ell^2$ the Hilbert space of (complex valued) square-summable sequences and denote $\ell^2(H)$ the Hilbert space of $H$-valued (norm)square-summable sequences.
Questions:
1) Do we have that $\ell^2(A)$ is a dense subspace of $\ell^2(H)$ ? (Here $\ell^2(A)$ is the natural subspace of $\ell^2(H)$ with $A$-valued sequences).
2) Do we have $\ell^2(A) \simeq \ell^2\otimes A$ (here $\otimes$ is the (algebraic) vector space tensor product). Does this (possible) isomorphism extends to the prehilbertian structures ?
3) Do we have $\ell^2(H) = \ell^2\widehat{\otimes} H$ (as tensor product of Hilbert space) ?
4) Do we have $\ell^2\otimes A$ dense in $\ell^2\widehat{\otimes} H$ ?