I have two questions which are intuitively true.


Let $V$ be a Hilbert space. As usual we can turn $V\otimes V$ or $V\otimes V\otimes V$ into Hilbert spaces by intorducing the natural inner product and by performing completion.


Question 1. We have a sequence of simple tensors $f\_{i}\otimes g\_{i}$ that converges in the Hilbert space $V\otimes V$ to some tensor. Is it true that the limit is a simple tensor, that is can be represented by $f\otimes g$? The matter is that $f\_{i}$ and $g\_{i}$ need not have limits, as the simple example $i f\otimes\frac{1}{i} g$, for some fixed $f,g\in V$ shows.


Question 2. If a sequence of simple tensors of the form $f\_{i}\otimes f\_{i}\otimes g\_{i}$ has a limit which is a simple tensor, can it be represented by $f\otimes f\otimes g$, for some $f,g\in V$?



Limit of simple tensors
Under the natural identification of the completion of $V\otimes W$ with Hilbert-Schmidt operators $V\rightarrow W^\*$, monomial tensors give rank-one operators. A Hilbert-Schmidt-norm limit of rank-one operators is certainly rank-one. (This viewpoint gets away from the pitfalls of specific representations of the tensors.) A similar argument works for $V\otimes V\otimes V$.


