Hilbert tensor product ($H{\otimes }_{2}K$)

The Hilbert tensor product is the completion of the algebraic tensor product of Hilbert spaces with $⟨u\otimes v,u^{\prime }\otimes v^{\prime }⟩=⟨u,u^{\prime }⟩⟨v,v^{\prime }⟩$, extended sesquilinearly. Orthonormal bases identify this inner product with the Euclidean inner product on coefficient arrays, proving positivity. In particular, $⟨u^{\otimes m},v^{\otimes m}⟩=⟨u,v{⟩}^m$ and $\Vert u^{\otimes m}\Vert =\Vert u{\Vert }^m$. These identities turn scalar power series into vector embeddings, as in the proof of the Grothendieck inequality.