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?
Best Answer
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:
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.