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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 $\ell^2\widehat{\otimes} H$ ?

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1 Answer 1

up vote 3 down vote accepted

1) If $n\geq1$, it is clear that the closure of $\ell^2(A)$ contains the subspace of $\ell^2(H)$ of those sequences which vanish from the $n$th component onwards. It then contains the closure of the union of these subspaces, which is $\ell^2(H)$.

2) Nope, in general. If $(a_n)_{n\geq1}$ is a linearly independent sequence of elements of $A$ such which is in $\ell^(A)$, then it is not in the subspace $\ell^2\otimes A\subseteq\ell^2(A)$.

3 and 4) Yes. Proceed as in (1) to show this.

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