= Hilbert tensor product
{c}
{title2=$H\otimes_2 K$}
The Hilbert tensor product is the completion of the algebraic <tensor product> of <Hilbert spaces> with $\langle u\otimes v,u'\otimes v'\rangle=\langle u,u'\rangle\langle v,v'\rangle$, extended sesquilinearly. Orthonormal bases identify this inner product with the Euclidean inner product on coefficient arrays, proving positivity. In particular, $\langle u^{\otimes m},v^{\otimes m}\rangle=\langle u,v\rangle^m$ and $\|u^{\otimes m}\|=\|u\|^m$. These identities turn scalar <power series> into vector embeddings, as in the proof of the <Grothendieck inequality>.
Back to article page
