In Cohen-Tannoudji's Quantum Mechanics book the tensor product of two two Hilbert spaces $(\mathcal H = \mathcal H_1 \otimes \mathcal H_2)$ was introduced in (2.312) by saying that to every pair of vectors $$|\phi(1)\rangle \in \mathcal H_1, |\chi(2)\rangle \in \mathcal H_2$$ there belongs a vector $$|\phi(1)\rangle \otimes |\chi(2)\rangle \in \mathcal H$$ In a footnote it stated that the order doesn't matter and that we could also call it $$|\chi(2)\rangle \otimes |\phi(1)\rangle$$ I'm a bit confused, since I though that the order of the tensor product generally matters. What would that expression look like if we picked a basis, say: $$|\phi(1)\rangle = a_1|u_1\rangle + a_2|u_2\rangle + \dotsc$$ $$|\chi(2)\rangle = b_1|v_1\rangle + b_2|v_2\rangle + \dotsc$$
Any help will be appreciated!