Hilbert space: product and tensor product space Let $H_1$ and $H_2$ be Hilbert spaces, then I would intuitively define the inner product on $H_1 \times H_2$ by $\langle (x_1,x_2),(y_1,y_2)\rangle = \langle x_1,y_1 \rangle + \langle x_2,y_2 \rangle$.
Now the inner product of the tensor space is given by $\langle (x_1 \otimes x_2),(y_1 \otimes y_2)\rangle = \langle x_1 y_1 \rangle \cdot \langle x_2,y_2 \rangle$ and I was wondering how this fits together?
Could anybody comment on the issue why we have to define the inner products exactly in that way for the two different spaces?

The way the cartesian and tensor product combine can be seen in the isomorphism of complex valued functions $$L^{2}(R\times R)\cong L^{2}(R)\otimes L^{2}(R)$$  The inner product on the left hand side is a double integral while that on the right is your product of inner products (integrals in one variable) . This identifies functions on $R\times R$ with limits of sums of functions of the form f(x)g(y) with respect to the norm generated by the appropriate inner product.
