• Images
• Videos
• Documents
• Books
• Articles

Definitions

Let X {\displaystyle X} and Y {\displaystyle Y} be locally convex topological vector spaces. Their projective tensor product X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} is the unique locally convex topological vector space with underlying vector space X ⊗ Y {\displaystyle X\otimes Y} having the following universal property:¹

For any locally convex topological vector space Z {\displaystyle Z} , if Φ Z {\displaystyle \Phi _{Z}} is the canonical map from the vector space of bilinear maps X × Y → Z {\displaystyle X\times Y\to Z} to the vector space of linear maps X ⊗ Y → Z {\displaystyle X\otimes Y\to Z} , then the image of the restriction of Φ Z {\displaystyle \Phi _{Z}} to the continuous bilinear maps is the space of continuous linear maps X ⊗ π Y → Z {\displaystyle X\otimes _{\pi }Y\to Z} .

When the topologies of X {\displaystyle X} and Y {\displaystyle Y} are induced by seminorms, the topology of X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} is induced by seminorms constructed from those on X {\displaystyle X} and Y {\displaystyle Y} as follows. If p {\displaystyle p} is a seminorm on X {\displaystyle X} , and q {\displaystyle q} is a seminorm on Y {\displaystyle Y} , define their tensor product p ⊗ q {\displaystyle p\otimes q} to be the seminorm on X ⊗ Y {\displaystyle X\otimes Y} given by ( p ⊗ q ) ( b ) = inf r > 0 , b ∈ r W r {\displaystyle (p\otimes q)(b)=\inf _{r>0,\,b\in rW}r} for all b {\displaystyle b} in X ⊗ Y {\displaystyle X\otimes Y} , where W {\displaystyle W} is the balanced convex hull of the set { x ⊗ y : p ( x ) ≤ 1 , q ( y ) ≤ 1 } {\displaystyle \left\{x\otimes y:p(x)\leq 1,q(y)\leq 1\right\}} . The projective topology on X ⊗ Y {\displaystyle X\otimes Y} is generated by the collection of such tensor products of the seminorms on X {\displaystyle X} and Y {\displaystyle Y} .²³ When X {\displaystyle X} and Y {\displaystyle Y} are normed spaces, this definition applied to the norms on X {\displaystyle X} and Y {\displaystyle Y} gives a norm, called the projective norm, on X ⊗ Y {\displaystyle X\otimes Y} which generates the projective topology.⁴

Properties

Throughout, all spaces are assumed to be locally convex. The symbol X ⊗ ^ π Y {\displaystyle X{\widehat {\otimes }}_{\pi }Y} denotes the completion of the projective tensor product of X {\displaystyle X} and Y {\displaystyle Y} .

• If X {\displaystyle X} and Y {\displaystyle Y} are both Hausdorff then so is X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} ;⁵ if X {\displaystyle X} and Y {\displaystyle Y} are Fréchet spaces then X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} is barelled.⁶
• For any two continuous linear operators u 1 : X 1 → Y 1 {\displaystyle u_{1}:X_{1}\to Y_{1}} and u 2 : X 2 → Y 2 {\displaystyle u_{2}:X_{2}\to Y_{2}} , their tensor product (as linear maps) u 1 ⊗ u 2 : X 1 ⊗ π X 2 → Y 1 ⊗ π Y 2 {\displaystyle u_{1}\otimes u_{2}:X_{1}\otimes _{\pi }X_{2}\to Y_{1}\otimes _{\pi }Y_{2}} is continuous.⁷
• In general, the projective tensor product does not respect subspaces (e.g. if Z {\displaystyle Z} is a vector subspace of X {\displaystyle X} then the TVS Z ⊗ π Y {\displaystyle Z\otimes _{\pi }Y} has in general a coarser topology than the subspace topology inherited from X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} ).⁸
• If E {\displaystyle E} and F {\displaystyle F} are complemented subspaces of X {\displaystyle X} and Y , {\displaystyle Y,} respectively, then E ⊗ F {\displaystyle E\otimes F} is a complemented vector subspace of X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} and the projective norm on E ⊗ π F {\displaystyle E\otimes _{\pi }F} is equivalent to the projective norm on X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} restricted to the subspace E ⊗ F {\displaystyle E\otimes F} . Furthermore, if X {\displaystyle X} and F {\displaystyle F} are complemented by projections of norm 1, then E ⊗ F {\displaystyle E\otimes F} is complemented by a projection of norm 1.⁹
• Let E {\displaystyle E} and F {\displaystyle F} be vector subspaces of the Banach spaces X {\displaystyle X} and Y {\displaystyle Y} , respectively. Then E ⊗ ^ F {\displaystyle E{\widehat {\otimes }}F} is a TVS-subspace of X ⊗ ^ π Y {\displaystyle X{\widehat {\otimes }}_{\pi }Y} if and only if every bounded bilinear form on E × F {\displaystyle E\times F} extends to a continuous bilinear form on X × Y {\displaystyle X\times Y} with the same norm.¹⁰

Completion

In general, the space X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} is not complete, even if both X {\displaystyle X} and Y {\displaystyle Y} are complete (in fact, if X {\displaystyle X} and Y {\displaystyle Y} are both infinite-dimensional Banach spaces then X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} is necessarily not complete¹¹). However, X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by X ⊗ ^ π Y {\displaystyle X{\widehat {\otimes }}_{\pi }Y} .

The continuous dual space of X ⊗ ^ π Y {\displaystyle X{\widehat {\otimes }}_{\pi }Y} is the same as that of X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} , namely, the space of continuous bilinear forms B ( X , Y ) {\displaystyle B(X,Y)} .¹²

Grothendieck's representation of elements in the completion

In a Hausdorff locally convex space X , {\displaystyle X,} a sequence ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} in X {\displaystyle X} is absolutely convergent if ∑ i = 1 ∞ p ( x i ) < ∞ {\displaystyle \sum _{i=1}^{\infty }p\left(x_{i}\right)<\infty } for every continuous seminorm p {\displaystyle p} on X . {\displaystyle X.} ¹³ We write x = ∑ i = 1 ∞ x i {\displaystyle x=\sum _{i=1}^{\infty }x_{i}} if the sequence of partial sums ( ∑ i = 1 n x i ) n = 1 ∞ {\displaystyle \left(\sum _{i=1}^{n}x_{i}\right)_{n=1}^{\infty }} converges to x {\displaystyle x} in X . {\displaystyle X.} ¹⁴

The following fundamental result in the theory of topological tensor products is due to Alexander Grothendieck.¹⁵

Theorem—Let X {\displaystyle X} and Y {\displaystyle Y} be metrizable locally convex TVSs and let z ∈ X ⊗ ^ π Y . {\displaystyle z\in X{\widehat {\otimes }}_{\pi }Y.} Then z {\displaystyle z} is the sum of an absolutely convergent series z = ∑ i = 1 ∞ λ i x i ⊗ y i {\displaystyle z=\sum _{i=1}^{\infty }\lambda _{i}x_{i}\otimes y_{i}} where ∑ i = 1 ∞ | λ i | < ∞ , {\displaystyle \sum _{i=1}^{\infty }|\lambda _{i}|<\infty ,} and ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} and ( y i ) i = 1 ∞ {\displaystyle \left(y_{i}\right)_{i=1}^{\infty }} are null sequences in X {\displaystyle X} and Y , {\displaystyle Y,} respectively.

The next theorem shows that it is possible to make the representation of z {\displaystyle z} independent of the sequences ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} and ( y i ) i = 1 ∞ . {\displaystyle \left(y_{i}\right)_{i=1}^{\infty }.}

Theorem¹⁶—Let X {\displaystyle X} and Y {\displaystyle Y} be Fréchet spaces and let U {\displaystyle U} (resp. V {\displaystyle V} ) be a balanced open neighborhood of the origin in X {\displaystyle X} (resp. in Y {\displaystyle Y} ). Let K 0 {\displaystyle K_{0}} be a compact subset of the convex balanced hull of U ⊗ V := { u ⊗ v : u ∈ U , v ∈ V } . {\displaystyle U\otimes V:=\{u\otimes v:u\in U,v\in V\}.} There exists a compact subset K 1 {\displaystyle K_{1}} of the unit ball in ℓ 1 {\displaystyle \ell ^{1}} and sequences ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} and ( y i ) i = 1 ∞ {\displaystyle \left(y_{i}\right)_{i=1}^{\infty }} contained in U {\displaystyle U} and V , {\displaystyle V,} respectively, converging to the origin such that for every z ∈ K 0 {\displaystyle z\in K_{0}} there exists some ( λ i ) i = 1 ∞ ∈ K 1 {\displaystyle \left(\lambda _{i}\right)_{i=1}^{\infty }\in K_{1}} such that z = ∑ i = 1 ∞ λ i x i ⊗ y i . {\displaystyle z=\sum _{i=1}^{\infty }\lambda _{i}x_{i}\otimes y_{i}.}

Topology of bi-bounded convergence

Let B X {\displaystyle {\mathfrak {B}}_{X}} and B Y {\displaystyle {\mathfrak {B}}_{Y}} denote the families of all bounded subsets of X {\displaystyle X} and Y , {\displaystyle Y,} respectively. Since the continuous dual space of X ⊗ ^ π Y {\displaystyle X{\widehat {\otimes }}_{\pi }Y} is the space of continuous bilinear forms B ( X , Y ) , {\displaystyle B(X,Y),} we can place on B ( X , Y ) {\displaystyle B(X,Y)} the topology of uniform convergence on sets in B X × B Y , {\displaystyle {\mathfrak {B}}_{X}\times {\mathfrak {B}}_{Y},} which is also called the topology of bi-bounded convergence. This topology is coarser than the strong topology on B ( X , Y ) {\displaystyle B(X,Y)} , 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 B ⊆ X ⊗ ^ Y , {\displaystyle B\subseteq X{\widehat {\otimes }}Y,} do there exist bounded subsets B 1 ⊆ X {\displaystyle B_{1}\subseteq X} and B 2 ⊆ Y {\displaystyle B_{2}\subseteq Y} such that B {\displaystyle B} is a subset of the closed convex hull of B 1 ⊗ B 2 := { b 1 ⊗ b 2 : b 1 ∈ B 1 , b 2 ∈ B 2 } {\displaystyle B_{1}\otimes B_{2}:=\{b_{1}\otimes b_{2}:b_{1}\in B_{1},b_{2}\in B_{2}\}} ?

Grothendieck proved that these topologies are equal when X {\displaystyle X} and Y {\displaystyle Y} are both Banach spaces or both are DF-spaces (a class of spaces introduced by Grothendieck¹⁷). They are also equal when both spaces are Fréchet with one of them being nuclear.¹⁸

Strong dual and bidual

Let X {\displaystyle X} be a locally convex topological vector space and let X ′ {\displaystyle X^{\prime }} be its continuous dual space. Alexander Grothendieck characterized the strong dual and bidual for certain situations:

Theorem¹⁹ (Grothendieck)—Let N {\displaystyle N} and Y {\displaystyle Y} be locally convex topological vector spaces with N {\displaystyle N} nuclear. Assume that both N {\displaystyle N} and Y {\displaystyle Y} are Fréchet spaces, or else that they are both DF-spaces. Then, denoting strong dual spaces with a subscripted b {\displaystyle b} :

1. The strong dual of N ⊗ ^ π Y {\displaystyle N{\widehat {\otimes }}_{\pi }Y} can be identified with N b ′ ⊗ ^ π Y b ′ {\displaystyle N_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime }} ;
2. The bidual of N ⊗ ^ π Y {\displaystyle N{\widehat {\otimes }}_{\pi }Y} can be identified with N ⊗ ^ π Y ′ ′ {\displaystyle N{\widehat {\otimes }}_{\pi }Y^{\prime \prime }} ;
3. If Y {\displaystyle Y} is reflexive then N ⊗ ^ π Y {\displaystyle N{\widehat {\otimes }}_{\pi }Y} (and hence N b ′ ⊗ ^ π Y b ′ {\displaystyle N_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime }} ) is a reflexive space;
4. Every separately continuous bilinear form on N b ′ × Y b ′ {\displaystyle N_{b}^{\prime }\times Y_{b}^{\prime }} is continuous;
5. Let L ( X b ′ , Y ) {\displaystyle L\left(X_{b}^{\prime },Y\right)} be the space of bounded linear maps from X b ′ {\displaystyle X_{b}^{\prime }} to Y {\displaystyle Y} . Then, its strong dual can be identified with N b ′ ⊗ ^ π Y b ′ , {\displaystyle N_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime },} so in particular if Y {\displaystyle Y} is reflexive then so is L b ( X b ′ , Y ) . {\displaystyle L_{b}\left(X_{b}^{\prime },Y\right).}

Examples

• For ( X , A , μ ) {\displaystyle (X,{\mathcal {A}},\mu )} a measure space, let L 1 {\displaystyle L^{1}} be the real Lebesgue space L 1 ( μ ) {\displaystyle L^{1}(\mu )} ; let E {\displaystyle E} be a real Banach space. Let L E 1 {\displaystyle L_{E}^{1}} be the completion of the space of simple functions X → E {\displaystyle X\to E} , modulo the subspace of functions X → E {\displaystyle X\to E} whose pointwise norms, considered as functions X → R {\displaystyle X\to \mathbb {R} } , have integral 0 {\displaystyle 0} with respect to μ {\displaystyle \mu } . Then L E 1 {\displaystyle L_{E}^{1}} is isometrically isomorphic to L 1 ⊗ ^ π E {\displaystyle L^{1}{\widehat {\otimes }}_{\pi }E} .²⁰

See also

• Inductive tensor product – binary operation on topological vector spacesPages displaying wikidata descriptions as a fallback
• Injective tensor product
• Tensor product of Hilbert spaces – Tensor product space endowed with a special inner product

Citations

• 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

• 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. MR 0075539. OCLC 9308061.
• 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.

External links

• Nuclear space at ncatlab

References

1. Trèves 2006, p. 438. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. https://search.worldcat.org/oclc/853623322 ↩

2. Trèves 2006, p. 435. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. https://search.worldcat.org/oclc/853623322 ↩

3. Trèves 2006, p. 438. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. https://search.worldcat.org/oclc/853623322 ↩

4. Trèves 2006, p. 437. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. https://search.worldcat.org/oclc/853623322 ↩

5. Trèves 2006, p. 437. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. https://search.worldcat.org/oclc/853623322 ↩

6. Trèves 2006, p. 445. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. https://search.worldcat.org/oclc/853623322 ↩

7. Trèves 2006, p. 439. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. https://search.worldcat.org/oclc/853623322 ↩

8. Ryan 2002, p. 18. - Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184. https://search.worldcat.org/oclc/48092184 ↩

9. Ryan 2002, p. 18. - Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184. https://search.worldcat.org/oclc/48092184 ↩

10. Ryan 2002, p. 24. - Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184. https://search.worldcat.org/oclc/48092184 ↩

11. Ryan 2002, p. 43. - Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184. https://search.worldcat.org/oclc/48092184 ↩

12. Schaefer & Wolff 1999, p. 173. - 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. https://search.worldcat.org/oclc/840278135 ↩

13. Schaefer & Wolff 1999, p. 120. - 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. https://search.worldcat.org/oclc/840278135 ↩

14. Schaefer & Wolff 1999, p. 120. - 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. https://search.worldcat.org/oclc/840278135 ↩

15. Schaefer & Wolff 1999, p. 94. - 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. https://search.worldcat.org/oclc/840278135 ↩

16. Trèves 2006, pp. 459–460. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. https://search.worldcat.org/oclc/853623322 ↩

17. Schaefer & Wolff 1999, p. 154. - 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. https://search.worldcat.org/oclc/840278135 ↩

18. Schaefer & Wolff 1999, p. 173. - 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. https://search.worldcat.org/oclc/840278135 ↩

19. Schaefer & Wolff 1999, pp. 175–176. - 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. https://search.worldcat.org/oclc/840278135 ↩

20. Schaefer & Wolff 1999, p. 95. - 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. https://search.worldcat.org/oclc/840278135 ↩