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The following question is probably too elementary and/or well-known for MathOverflow, so I'll try here:

Let $ell^2 mathbin{hatotimes_pi} ell^2$ and $ell^2 mathbin{hatotimes_varepsilon} ell^2$ refer to the (completed) projective and injective tensor products (as defined, say, in Wikipedia), as Banach spaces, of the Hilbert space $ell^2 = {ucolonmathbb{N}tomathbb{R} : sum_{k=0}^{+infty} u_k^2 < +infty}$ of square-summable sequences with itself.

I understand that it is not easy to describe these spaces, but I wonder if it is still possible to give a reasonably concrete condition for a “sequence of sequences” (i.e., a function $mathbb{N}^2 to mathbb{R}$) to belong to one or the other?

More precisely, if we consider the continuous linear form $e_k^*colonell^2tomathbb{R}$ which maps $u in ell^2$ to its $k$-th term $langle u, e_krangle$, then the tensor product $e_m^* otimes e_n^*$ defines a continuous linear form of norm $1$ on either $ell^2 mathbin{hatotimes_pi} ell^2$ or $ell^2 mathbin{hatotimes_varepsilon} ell^2$, so an element $v$ in one of these spaces defines an array $J(v)colon (m,n) mapsto (e_m^* otimes e_n^*)(v)$, which belongs to $ell^infty(mathbb{N}^2)$ (the space of bounded functions $mathbb{N}^2tomathbb{R}$). This, in turn, defines a continuous linear map $J_alpha colon ell^2 mathbin{hatotimes_alpha} ell^2 to ell^infty(mathbb{N}^2)$ (of norm $1$) for $alpha in {pi,varepsilon}$. I guess I have four questions:

• Is $J_pi$ injective? (Can $ell^2 mathbin{hatotimes_pi} ell^2$ be seen as a space of functions $mathbb{N}^2tomathbb{R}$?)




• Is $J_varepsilon$ injective? (Can $ell^2 mathbin{hatotimes_varepsilon} ell^2$ be seen as a space of functions $mathbb{N}^2tomathbb{R}$?)




• What is the image of $J_pi$? (When does a function $mathbb{N}^2tomathbb{R}$ belong to $ell^2 mathbin{hatotimes_pi} ell^2$?)




• What is the image of $J_varepsilon$? (When does a function $mathbb{N}^2tomathbb{R}$ belong to $ell^2 mathbin{hatotimes_varepsilon} ell^2$?)

Here's a positive answer to the first two questions, and a very partial answer to the other two. Let the reference [R] stand for Ryan's book Introduction to Tensor Products of Banach Spaces (2002).

Let $mathcal{B}$ be the vector space of continous bilinear forms $ell^2 times ell^2 to mathbb{R}$ with the usual norm $|B| := sup{|B(x,y)| : |x|leq 1, |y|leq 1}$ for $B in mathcal{B}$. There is an obvious map $I colon mathcal{B} to ell^infty(mathbb{N}^2)$ taking $B$ to $(B(e_m,e_n))$; this map $I$ is injective and continuous of norm $1$ (injectivity of $I$ follows from the fact that the $e_n$ span a dense subset of $ell^2$; the fact that $|I|leq 1$ is clear from the definition of the norm on $mathcal{B}$; and that $|I|=1$ follows from the fact that $|I(e_m^*otimes e_n^*)| = 1$).

Now the maps $J_pi$ and $J_varepsilon$ defined in the question factor through this $Icolon mathcal{B} to ell^infty(mathbb{N}^2)$, the factors being the natural maps, which we also denote $J_pi$ and $J_varepsilon$, from $ell^2 mathbin{hatotimes_alpha} ell^2$ (for $alpha$ in ${pi,varepsilon}$) to $mathcal{B}$. So to answer the first two questions positively, it is enough to show the injectivity of these $J_alpha colon ell^2 mathbin{hatotimes_alpha} ell^2 to mathcal{B}$.

In the case of $J_varepsilon colon ell^2 mathbin{hatotimes_varepsilon} ell^2 to mathcal{B}$, injectivity is almost by definition: as explained in [R, §3.1], $J_varepsilon$ is norm preserving and we can see $ell^2 mathbin{hatotimes_varepsilon} ell^2$ as the closure of $ell^2 otimes ell^2$ inside $mathcal{B}$. Its image is the set of "approximable" bilinear forms.

In the case of $J_pi colon ell^2 mathbin{hatotimes_pi} ell^2 to mathcal{B}$, injectivity is not automatic (see [R, §2.6]). Its image is the set of "nuclear" bilinear forms. Nevertheless, since $ell^2$ is a Hilbert space, as a Banach space it has the approximation property ([R, ex. 4.4]), so by [R, cor. 4.8], the map $J_pi$ is indeed injective.

To summarize, $J_varepsilon$ is an isometric embedding which identifies $ell^2 mathbin{hatotimes_varepsilon} ell^2$ with the closure of $ell^2 otimes ell^2$ inside $mathcal{B}$ or "approximable" bilinear forms; as for $J_pi$, it is injective of norm $1$, and its image consists of "nuclear" bilinear forms.

It is also worth noting that $ell^2 mathbin{hatotimes_varepsilon} ell^2$ can also be identified with compact operators $ell^2 to ell^2$ ([R, prop. 4.12]), and that $ell^2 mathbin{hatotimes_pi} ell^2$ isometrically embeds in the dual of $ell^2 mathbin{hatotimes_varepsilon} ell^2$ ([R, thm. 4.14]).

I still don't have a clear picture of how one can tell whether an element of $mathcal{B}$, let alone $ell^infty(mathbb{N}^2)$, is approximable resp. nuclear, nor do I know if a concrete description can be given. I can give one example, though: the bilinear form corresponding to the Hilbert space dot product on $ell^2$, whose image by $Icolon mathcal{B} to ell^infty(mathbb{N}^2)$ takes the value $1$ on the diagonal and $0$ elsewhere, is in the image of $J_varepsilon$ but not of $J_pi$.