Tensoring a Hilbert Space by a Finite Dimensional Hilbert Space

Question

Given two Hilbert spaces $H_1$ and $H_2$ where $H_1$ is infinite dimensional and $H_2$ is finite dimensional, what is the easiest way to see that their tensor product, endowed with the tensor product inner product is already complete?

Answer 1

Let $\mathbb{F}$ be the base field ($\mathbb{R}$ or $\mathbb{C}$). Then if $\dim(H_2) = n,$ we have that $H_2\simeq \mathbb{F}^n = \mathbb{F}\oplus\cdots\oplus\mathbb{F}$. For Hilbert spaces, in fact it's true much more generally, tensor product distributes over direct sums.

Thus we have that $H_1\otimes H_2\simeq H_1\oplus\cdots\oplus H_1$. Note that we haven't taken any completions. So now it just reduces to the fact that the direct sum of two Hilbert spaces is a Hilbert space, again without need for take any completions.