Editing Tensor product (section)

=== Topological tensor products ===
{{main|Topological tensor product|Tensor product of Hilbert spaces}}

[[Hilbert space]]s generalize finite-dimensional vector spaces to arbitrary dimensions. There is [[tensor product of Hilbert spaces|an analogous operation]], also called the "tensor product," that makes Hilbert spaces a [[symmetric monoidal category]].  It is essentially constructed as the [[Complete_metric_space#Completion|metric space completion]] of the algebraic tensor product discussed above.  However, it does not satisfy the obvious analogue of the universal property defining tensor products;<ref>{{cite web|url=https://www-users.cse.umn.edu/~garrett/m/v/nonexistence_tensors.pdf|date=July 22, 2010|title=Non-existence of tensor products of Hilbert spaces|first=Paul|last=Garrett}}</ref> the morphisms for that property must be restricted to [[Hilbert–Schmidt operator]]s.<ref>{{Cite book | last1=Kadison | first1=Richard&nbsp;V. | last2=Ringrose | first2=John R. | title=Fundamentals of the theory of operator algebras | volume=I | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=[[Graduate Studies in Mathematics]] | isbn=978-0-8218-0819-1 | mr= 1468229 | year=1997 | at=Thm.&nbsp;2.6.4.}}</ref>

In situations where the imposition of an inner product is inappropriate, one can still attempt to complete the algebraic tensor product, as a [[topological tensor product]].  However, such a construction is no longer uniquely specified: in many cases, there are multiple natural topologies on the algebraic tensor product.