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 $V\otimes V\otimes V$.

It is perhaps worth putting on record that this is a special case of an elementary result. A function $f$ on the product $X \times Y$ of two abstract sets is a simple tensor if and only if it is the product $f(x,y) = g(x)h(y)$ of two functions of one variable. Since this can be expressed in the pointwise condition $f(x_1,y_1)f(x_0,y_0)=f(x_1,y_0)f(x_0,y_1)$ for all suitable pairs, the class of simple tensors is closed in most useful topologies. The versions of this result which are useful in linear algebra or functional analysis can be obtained from this one by using the standard method of regarding vectors as functions on the dual space. If $f$ is continuous, then so are $g$ and $h$. If it is bilinear, then $g$ and $h$ are linear.