Editing Tensor (section)

=== Tensors in infinite dimensions ===
The notion of a tensor can be generalized in a variety of ways to [[Dimension (vector space)|infinite dimensions]].  One, for instance, is via the [[tensor product of Hilbert spaces|tensor product]] of [[Hilbert space]]s.<ref>{{cite journal | last1 = Segal | first1 = I. E. | date=January 1956 | title = Tensor Algebras Over Hilbert Spaces. I | journal = [[Transactions of the American Mathematical Society]] | volume = 81 | issue = 1 | pages = 106–134 | jstor = 1992855 | doi = 10.2307/1992855 | doi-access = free }}</ref>  Another way of generalizing the idea of tensor, common in [[Nonlinear system|nonlinear analysis]], is via the [[#As multilinear maps|multilinear maps definition]] where instead of using finite-dimensional vector spaces and their [[algebraic dual]]s, one uses infinite-dimensional [[Banach space]]s and their [[continuous dual]].<ref>{{cite book
|last1=Abraham |first1=Ralph
|last2=Marsden |first2=Jerrold E.
|last3=Ratiu |first3=Tudor S.
|chapter-url={{google books |plainurl=y |id=dWHet_zgyCAC}}
|title=Manifolds, Tensor Analysis and Applications
|edition=2nd |series=Applied Mathematical Sciences |volume=75
|date= February 1988
|publisher=Springer
|isbn=978-0-387-96790-5
|oclc= 18562688
|pages=338–9
|chapter=5. Tensors
|quote=Elements of T<sup>r</sup><sub>s</sub> are called tensors on E, [...].
}}</ref>  Tensors thus live naturally on [[Banach manifold]]s<ref>{{cite book | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Differential manifolds |url={{google books |plainurl=y |id=dn7rBwAAQBAJ}}| publisher=[[Addison-Wesley]] | year=1972 |isbn= 978-0-201-04166-8 }}</ref> and [[Fréchet manifold]]s.