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Projective tensor product

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In functional analysis, an area of mathematics, the projective tensor product of two locally convex topological vector spaces is a natural topological vector space structure on their tensor product. Namely, given locally convex topological vector spaces and , the projective topology, or π-topology, on is the strongest topology which makes a locally convex topological vector space such that the canonical map (from to ) is continuous. When equipped with this topology, is denoted and called the projective tensor product of and .

Definitions

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Let and be locally convex topological vector spaces. Their projective tensor product is the unique locally convex topological vector space with underlying vector space having the following universal property:[1]

For any locally convex topological vector space , if is the canonical map from the vector space of bilinear maps to the vector space of linear maps , then the image of the restriction of to the continuous bilinear maps is the space of continuous linear maps .

When the topologies of and are induced by seminorms, the topology of is induced by seminorms constructed from those on and as follows. If is a seminorm on , and is a seminorm on , define their tensor product to be the seminorm on given by for all in , where is the balanced convex hull of the set . The projective topology on is generated by the collection of such tensor products of the seminorms on and .[2][1] When and are normed spaces, this definition applied to the norms on and gives a norm, called the projective norm, on which generates the projective topology.[3]

Properties

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Throughout, all spaces are assumed to be locally convex. The symbol denotes the completion of the projective tensor product of and .

  • If and are both Hausdorff then so is ;[3] if and are Fréchet spaces then is barelled.[4]
  • For any two continuous linear operators and , their tensor product (as linear maps) is continuous.[5]
  • In general, the projective tensor product does not respect subspaces (e.g. if is a vector subspace of then the TVS has in general a coarser topology than the subspace topology inherited from ).[6]
  • If and are complemented subspaces of and respectively, then is a complemented vector subspace of and the projective norm on is equivalent to the projective norm on restricted to the subspace . Furthermore, if and are complemented by projections of norm 1, then is complemented by a projection of norm 1.[6]
  • Let and be vector subspaces of the Banach spaces and , respectively. Then is a TVS-subspace of if and only if every bounded bilinear form on extends to a continuous bilinear form on with the same norm.[7]

Completion

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In general, the space is not complete, even if both and are complete (in fact, if and are both infinite-dimensional Banach spaces then is necessarily not complete[8]). However, can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by .

The continuous dual space of is the same as that of , namely, the space of continuous bilinear forms .[9]

Grothendieck's representation of elements in the completion

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In a Hausdorff locally convex space a sequence in is absolutely convergent if for every continuous seminorm on [10] We write if the sequence of partial sums converges to in [10]

The following fundamental result in the theory of topological tensor products is due to Alexander Grothendieck.[11]

Theorem — Let and be metrizable locally convex TVSs and let Then is the sum of an absolutely convergent series where and and are null sequences in and respectively.

The next theorem shows that it is possible to make the representation of independent of the sequences and

Theorem[12] — Let and be Fréchet spaces and let (resp. ) be a balanced open neighborhood of the origin in (resp. in ). Let be a compact subset of the convex balanced hull of There exists a compact subset of the unit ball in and sequences and contained in and respectively, converging to the origin such that for every there exists some such that

Topology of bi-bounded convergence

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Let and denote the families of all bounded subsets of and respectively. Since the continuous dual space of is the space of continuous bilinear forms we can place on the topology of uniform convergence on sets in which is also called the topology of bi-bounded convergence. This topology is coarser than the strong topology on , and in (Grothendieck 1955), Alexander Grothendieck was interested in when these two topologies were identical. This is equivalent to the problem: Given a bounded subset do there exist bounded subsets and such that is a subset of the closed convex hull of ?

Grothendieck proved that these topologies are equal when and are both Banach spaces or both are DF-spaces (a class of spaces introduced by Grothendieck[13]). They are also equal when both spaces are Fréchet with one of them being nuclear.[9]

Strong dual and bidual

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Let be a locally convex topological vector space and let be its continuous dual space. Alexander Grothendieck characterized the strong dual and bidual for certain situations:

Theorem[14] (Grothendieck) — Let and be locally convex topological vector spaces with nuclear. Assume that both and are Fréchet spaces, or else that they are both DF-spaces. Then, denoting strong dual spaces with a subscripted :

  1. The strong dual of can be identified with ;
  2. The bidual of can be identified with ;
  3. If is reflexive then (and hence ) is a reflexive space;
  4. Every separately continuous bilinear form on is continuous;
  5. Let be the space of bounded linear maps from to . Then, its strong dual can be identified with so in particular if is reflexive then so is

Examples

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  • For a measure space, let be the real Lebesgue space ; let be a real Banach space. Let be the completion of the space of simple functions , modulo the subspace of functions whose pointwise norms, considered as functions , have integral with respect to . Then is isometrically isomorphic to .[15]

See also

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Citations

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  1. ^ a b Trèves 2006, p. 438.
  2. ^ Trèves 2006, p. 435.
  3. ^ a b Trèves 2006, p. 437.
  4. ^ Trèves 2006, p. 445.
  5. ^ Trèves 2006, p. 439.
  6. ^ a b Ryan 2002, p. 18.
  7. ^ Ryan 2002, p. 24.
  8. ^ Ryan 2002, p. 43.
  9. ^ a b Schaefer & Wolff 1999, p. 173.
  10. ^ a b Schaefer & Wolff 1999, p. 120.
  11. ^ Schaefer & Wolff 1999, p. 94.
  12. ^ Trèves 2006, pp. 459–460.
  13. ^ Schaefer & Wolff 1999, p. 154.
  14. ^ Schaefer & Wolff 1999, pp. 175–176.
  15. ^ Schaefer & Wolff 1999, p. 95.

References

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  • Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

Further reading

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  • Diestel, Joe (2008). The metric theory of tensor products : Grothendieck's résumé revisited. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-4440-3. OCLC 185095773.
  • Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). 16. Providence: American Mathematical Society. ISBN 978-0-8218-1216-7. MR 0075539. OCLC 1315788.
  • Grothendieck, Grothendieck (1966). Produits tensoriels topologiques et espaces nucléaires (in French). Providence: American Mathematical Society. ISBN 0-8218-1216-5. OCLC 1315788.
  • Pietsch, Albrecht (1972). Nuclear locally convex spaces. Berlin, New York: Springer-Verlag. ISBN 0-387-05644-0. OCLC 539541.
  • Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.
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