Let $H_1$ and $H_2$ be finite-dimensional Hilbert spaces with inner products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ respectively. Construct the tensor product of $H_1$ and $H_2$ as $H_1\otimes H_2$, then we can define an inner product by setting $\langle u_1\otimes u_2,v_1\otimes v_2\rangle:=\langle u_1,v_1\rangle_1\langle u_2,v_2\rangle_2$ for $u_1,v_1\in H_1,u_2,v_2\in H_2$. This is how all textbooks define it.
My question is: Can we write the sum $u_1\otimes u_2+x_1\otimes x_2$ as one pure tensor product and consider something like $\langle u_1\otimes u_2+x_1\otimes x_2,v_1⊗v_2\rangle$?