Tensor product of Hilbert spaces

Summary

The tensor product of two Hilbert spaces is the completion of their algebraic tensor product. Its inner product multiplies the inner products of the factors. It is again a Hilbert space, and products of orthonormal basis vectors form an orthonormal basis of it. In quantum mechanics it describes a system made of two subsystems. In analysis, $L^2(X) \mathbin{\widehat{\otimes}} L^2(Y)$ is $L^2(X \times Y)$.