Tensor product of Hilbert spaces

In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category. In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category. Since Hilbert spaces have inner products, one would like to introduce an inner product, and therefore a topology, on the tensor product that arise naturally from those of the factors. Let H1 and H2 be two Hilbert spaces with inner products ⟨ ⋅ , ⋅ ⟩ 1 {displaystyle langle cdot ,cdot angle _{1}} and ⟨ ⋅ , ⋅ ⟩ 2 {displaystyle langle cdot ,cdot angle _{2}} , respectively. Construct the tensor product of H1 and H2 as vector spaces as explained in the article on tensor products. We can turn this vector space tensor product into an inner product space by defining and extending by linearity. That this inner product is the natural one is justified by the identification of scalar-valued bilinear maps on H1 × H2 and linear functionals on their vector space tensor product. Finally, take the completion under this inner product. The resulting Hilbert space is the tensor product of  H1 and H2. The tensor product can also be defined without appealing to the metric space completion. If H1 and H2 are two Hilbert spaces, one associates to every simple tensor product x 1 ⊗ x 2 {displaystyle x_{1}otimes x_{2}} the rank one operator from H 1 ∗ {displaystyle H_{1}^{*}} to H2 that maps a given x ∗ ∈ H 1 ∗ {displaystyle x^{*}in H_{1}^{*}} as This extends to a linear identification between H 1 ⊗ H 2 {displaystyle H_{1}otimes H_{2}} and the space of finite rank operators from H 1 ∗ {displaystyle H_{1}^{*}} to H2. The finite rank operators are embedded in the Hilbert space H S ( H 1 ∗ , H 2 ) {displaystyle HS(H_{1}^{*},H_{2})} of Hilbert–Schmidt operators from H 1 ∗ {displaystyle H_{1}^{*}} to H2. The scalar product in H S ( H 1 ∗ , H 2 ) {displaystyle HS(H_{1}^{*},H_{2})} is given by where ( e n ∗ ) {displaystyle (e_{n}^{*})} is an arbitrary orthonormal basis of H 1 ∗ . {displaystyle H_{1}^{*}.} Under the preceding identification, one can define the Hilbertian tensor product of H1 and H2, that is isometrically and linearly isomorphic to H S ( H 1 ∗ , H 2 ) . {displaystyle HS(H_{1}^{*},H_{2}).} The Hilbert tensor product H = H 1 ⊗ H 2 {displaystyle H=H_{1}otimes H_{2}} is characterized by the following universal property (Kadison & Ringrose 1983, Theorem 2.6.4): A weakly Hilbert-Schmidt mapping L : H1 × H2 → K is defined as a bilinear map for which a real number d exists, such that