Tensor Product: Denseness

Question

This is thread is just a note.

Given Hilbert spaces.

Then denseness will be inherited on tensor products: $$\mathcal{D},\mathcal{E}\text{ dense}\implies\mathcal{D}\otimes\mathcal{E}\text{ dense}$$

How to prove this from scratch without orthonormal bases?

Answer 1

This answer is community wiki.

First for simple tensors one has: $$\|\varphi\otimes\psi-\varphi_n\otimes\psi_n\|\leq\|\varphi-\varphi_n\|\cdot\|\psi\|+\|\varphi_n\|\cdot\|\psi-\psi_n\|\leq\delta_\varphi(1+\|\psi\|)+\|\varphi\|\delta_\psi$$ Next for sums it still holds true by continuity of addition: $$\|\varphi\otimes\psi+\varphi'\otimes\psi'\|\leq\|\varphi\otimes\psi\|+\|\varphi'\otimes\psi'\|$$ Finally the result follows as sums are dense in the Hilbert tensor product.