Editing Tensor (section)

=== Tensors in infinite dimensions ===
This discussion of tensors so far assumes finite dimensionality of the spaces involved, where the spaces of tensors obtained by each of these constructions are [[naturally isomorphic]].<ref group="Note">The [[Dual space#Injection into the double-dual|double duality isomorphism]], for instance, is used to identify ''V'' with the double dual space ''V''<sup>∗∗</sup>, which consists of multilinear forms of degree one on ''V''<sup>∗</sup>.  It is typical in linear algebra to identify spaces that are naturally isomorphic, treating them as the same space.</ref>  Constructions of spaces of tensors based on the tensor product and multilinear mappings can be generalized, essentially without modification, to [[vector bundle]]s or [[coherent sheaves]].<ref>{{cite book|first=N. |last=Bourbaki|title=Algebra I: Chapters 1-3|chapter=3|chapter-url={{google books |plainurl=y |id=STS9aZ6F204C}}|date=1998|publisher=Springer |isbn=978-3-540-64243-5}} where the case of finitely generated projective modules is treated.  The global sections of sections of a vector bundle over a compact space form a projective module over the ring of smooth functions.  All statements for coherent sheaves are true locally.</ref>  For infinite-dimensional vector spaces, inequivalent topologies lead to inequivalent notions of tensor, and these various isomorphisms may or may not hold depending on what exactly is meant by a tensor (see [[topological tensor product]]).  In some applications, it is the [[tensor product of Hilbert spaces]] that is intended, whose properties are the most similar to the finite-dimensional case. A more modern view is that it is the tensors' structure as a [[symmetric monoidal category]] that encodes their most important properties, rather than the specific models of those categories.<ref>{{citation|title= Braided tensor categories |first1= André |last1=Joyal |first2= Ross |last2=Street |journal= [[Advances in Mathematics]] |year=1993 |volume=102 |pages= 20–78 |doi= 10.1006/aima.1993.1055 |doi-access=free }}</ref>