The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category.
as vector spaces as explained in the article on tensor products.
That this inner product is the natural one is justified by the identification of scalar-valued bilinear maps on
and linear functionals on their vector space tensor product.
The tensor product can also be defined without appealing to the metric space completion.
are two Hilbert spaces, one associates to every simple tensor product
The finite rank operators are embedded in the Hilbert space
Under the preceding identification, one can define the Hilbertian tensor product of
As with any universal property, this characterizes the tensor product H uniquely, up to isomorphism.
The same universal property, with obvious modifications, also applies for the tensor product of any finite number of Hilbert spaces.
It is essentially the same universal property shared by all definitions of tensor products, irrespective of the spaces being tensored: this implies that any space with a tensor product is a symmetric monoidal category, and Hilbert spaces are a particular example thereof.
Two different definitions have historically been proposed for the tensor product of an arbitrary-sized collection
Von Neumann's traditional definition simply takes the "obvious" tensor product: to compute
The latter describes a pre-inner product through the polarization identity, so take the closed span of such simple tensors modulo that inner product's isotropy subspaces.
This definition is almost never separable, in part because, in physical applications, "most" of the space describes impossible states.
Modern authors typically use instead a definition due to Guichardet: to compute
in each Hilbert space, and then collect all simple tensors of the form
Then the von Neumann tensor product of the von Neumann algebras is the strong completion of the set of all finite linear combinations of simple tensor products
This is exactly equal to the von Neumann algebra of bounded operators of
Unlike for Hilbert spaces, one may take infinite tensor products of von Neumann algebras, and for that matter C*-algebras of operators, without defining reference states.
[3] This is one advantage of the "algebraic" method in quantum statistical mechanics.
The following examples show how tensor products arise naturally.
The definition of the product measure ensures that all functions of this form are square integrable, so this defines a bilinear mapping
It turns out that the set of linear combinations is in fact dense in
and it also explains why we need to take the completion in the construction of the Hilbert space tensor product.
Tensor products of Hilbert spaces arise often in quantum mechanics.
then the system consisting of both particles is described by the tensor product of
For example, the state space of a quantum harmonic oscillator is
Therefore, the two-particle system is described by wave functions of the form
A more intricate example is provided by the Fock spaces, which describe a variable number of particles.