Infinite product and tensor products

Question

Let $k$ be a field. Is it well-known that, in general, the natural inclusion map $$\Pi_{i=0}^\infty k \otimes_k \Pi_{i=0}^\infty k \rightarrow \Pi_{i,j=0}^\infty k$$ is not an isomorphism. However, clearly, this identifies the tensor products as a submodule of $\Pi_{i,j=0}^\infty k.$ How can this submodule be characterized?

Answer 1

Set $$ V:=\prod_{i,j=0}^\infty k $$ and let $W\subset V$ be the subspace in question.

Let $a:=(a_{ij})_{i,j=0}^\infty$ be in $V.$

Then $a$ is in $W$ if and only if the matrices $(a_{ij})_{i,j=0}^n$ have bounded ranks.

The condition is clearly necessary.

To prove that it is sufficient, assume that $a$ satisfies the condition. Set $$ a_{i\bullet}:=(a_{ij})_{j=0}^\infty. $$ For $n$ large enough, each $a_{i\bullet}$ is a linear combination of $a_{0\bullet},\dots,a_{n\bullet}$, say $$ a_{i\bullet}=b_{i0}a_{0\bullet}+\dots+b_{in}a_{n\bullet}, $$ and we get $$ a_{ij}=b_{i0}a_{0j}+\dots+b_{in}a_{nj} $$ for all $i,j$. This implies that $a$ is in $W$, as desired.