Distributivity of tensor product over a direct sum

Question

Let $\mathcal{H}, \mathcal{K}$ be finite dimensional Hilbert spaces and consider the space $$\left(\mathcal{H} \oplus \mathcal{K}\right) \otimes L^2(\mathbb{R}).$$ I would like a reference to show that this is isomorphic to $$\left(\mathcal{H} \otimes L^2(\mathbb{R})\right) \oplus \left(\mathcal{K} \otimes L^2(\mathbb{R})\right)$$ if it does indeed hold here. I've seen this result if all of the above were finite-dimensional vector spaces, but not for an infinite dimensional case, even though we have separability for $L^2(\mathbb{R})$. Thanks in advanced!

Answer 1

These are stated in Atiyah-Macdonald's Introduction to Commutative Algebra (pg 26) for general modules, assuming neither finite dimensionality (generation) nor that the underlying ring is a field.

Depending on your background, you could just check that the standard proof using the universal property goes through without any assumptions about finite-dimensionality.