Dense subspaces of a tensor product

Question

Let $\mathcal{H}$ be a Hilbert space (infinite dimensional, in general) then consider its tensor square: $\mathcal{H} \otimes \mathcal{H}$. Is the space $$\{f\otimes f \ | \ f\in \mathcal{H}\}$$ Dense in $\mathcal{H} \otimes \mathcal{H}$? If this is true then I suppose it is also true for arbitrary tensor powers of $\mathcal{H}$ also.

Answer 1

I'll talk about the subspace $\mathcal L \subset \mathcal H \otimes \mathcal H$ spanned by $\{f\otimes f\}$.

$\mathcal L$ is not dense in $\mathcal H \otimes \mathcal H$ unless $\dim \mathcal H \leq 1$.

Consider the operator on $\mathcal H \otimes \mathcal H$ defined as

$$A(x \otimes y) = x\otimes y - y \otimes x$$

and extended by linearity. $A$ is easily seen to be continuous; furthemore, $\forall v \in \mathcal L \quad A(v) = 0$.

However, if $x,y \in \mathcal H$ are linearly independent, $A(x \otimes y) \neq 0$. Thus, $\ker A$ is a proper closed subspace of $\mathcal H \otimes \mathcal H$ containing $\mathcal L$, so the latter cannot be dense.