Intuition behind canonical definition of inner product structure of tensors Consider Napkin Problem 13D (a):

Let $V$ and $W$ be finite-dimensional inner product spaces over $k$, where $k$ is either $\mathbb{R}$ or $\mathbb{C}$. Find a canonical way to make $V \otimes_k W$ into an inner product space too.

I approached this by working on special cases. Let $e_1, \dots , e_n$ and $f_1, \dots , f_m$ are orthonormal bases for $V$ and $W$, respectively.
I think we want $\langle e_1 \otimes f_1, e_1 \otimes f_1 \rangle_{V \otimes W} = 1$, since it feels like $\|e_1\| = \|f_1\| = 1$ should imply that $\| e_1 \otimes f_1 \|$. We also want $\langle e_1 \otimes f_1, e_2 \otimes f_2 \rangle_{V \otimes W} = 0$, since it feels like the fact that $e_1$ and $e_2$ are orthogonal and $f_1$ and $f_2$ are orthogonal should imply that $e_1 \otimes f_1$ and $e_2 \otimes f_2$ are orthogonal.
So for any $v_1, v_2 \in V$ and $w_1, w_2 \in W$, the inner form $\langle v_1 \otimes w_1, v_2 \otimes w_2 \rangle_{V \otimes W} := \langle v_1, v_2 \rangle_V \langle w_1, w_2 \rangle_W$ seems like a good choice, and it ended up being the intended inner form.
But I'm still very uncomfortable with the intuition behind this; is there a more solid way to understand why we take the product of the individual inner forms to get the inner form of the tensor product?
 A: Canonical is an imprecise word, but here is a justification. An inner product on $V$ is a map $V \otimes V \to k$ with certain properties. So having inner products on $V$ and $W$ gives a map
$$(V \otimes W) \otimes (V \otimes W) \cong (V \otimes V) \otimes (W \otimes W) \to k \otimes k \xrightarrow{(x \otimes y) \mapsto xy} k$$
and I would argue that the first isomorphism and the third map, the multiplication on $k$, are the canonical maps with those domains and targets. This composition is exactly the same as your formula, but all it says is that given bilinear forms on $V$ and $W$ there is a "canonical" bilinear form on $V \otimes W$. You still have to check that it's an inner product if you start with inner products.
To add some intuition in line with @Milten's comment, an inner product (or more generally a bilinear form) on $V$ is a product on $V$ taking values in $k$, so is a function on $V \otimes V$ since that's the universal target for a product on $V$. To have a way to multiply $v_1 \otimes w_1, v_2 \otimes w_2 \in V \otimes W$ (*) it should suffice to know how to:

*

*rearrange to $(v_1 \otimes v_2) \otimes (w_1 \otimes w_2)$;

*multiply $v_1$ with $v_2$ and $w_1$ with $w_2$ individually to get elements of $k$;

*multiply in $k$.

These are precisely the three maps that are being composed; 1 and 3 are "canonical" and 2 is the input.
(*) I'm ignoring that not all elements are pure tensors, but a "product" should satisfy distributive law so it's enough to know what it does to a generating set anyway.
