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Definition - Integral forms as the dual of the injective tensor product

See also: Injective tensor product and Projective tensor product

Let X and Y be locally convex TVSs, let X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} denote the projective tensor product, X ⊗ ^ π Y {\displaystyle X{\widehat {\otimes }}_{\pi }Y} denote its completion, let X ⊗ ϵ Y {\displaystyle X\otimes _{\epsilon }Y} denote the injective tensor product, and X ⊗ ^ ϵ Y {\displaystyle X{\widehat {\otimes }}_{\epsilon }Y} denote its completion. Suppose that In : X ⊗ ϵ Y → X ⊗ ^ ϵ Y {\displaystyle \operatorname {In} :X\otimes _{\epsilon }Y\to X{\widehat {\otimes }}_{\epsilon }Y} denotes the TVS-embedding of X ⊗ ϵ Y {\displaystyle X\otimes _{\epsilon }Y} into its completion and let t In : ( X ⊗ ^ ϵ Y ) b ′ → ( X ⊗ ϵ Y ) b ′ {\displaystyle {}^{t}\operatorname {In} :\left(X{\widehat {\otimes }}_{\epsilon }Y\right)_{b}^{\prime }\to \left(X\otimes _{\epsilon }Y\right)_{b}^{\prime }} be its transpose, which is a vector space-isomorphism. This identifies the continuous dual space of X ⊗ ϵ Y {\displaystyle X\otimes _{\epsilon }Y} as being identical to the continuous dual space of X ⊗ ^ ϵ Y {\displaystyle X{\widehat {\otimes }}_{\epsilon }Y} .

Let Id : X ⊗ π Y → X ⊗ ϵ Y {\displaystyle \operatorname {Id} :X\otimes _{\pi }Y\to X\otimes _{\epsilon }Y} denote the identity map and t Id : ( X ⊗ ϵ Y ) b ′ → ( X ⊗ π Y ) b ′ {\displaystyle {}^{t}\operatorname {Id} :\left(X\otimes _{\epsilon }Y\right)_{b}^{\prime }\to \left(X\otimes _{\pi }Y\right)_{b}^{\prime }} denote its transpose, which is a continuous injection. Recall that ( X ⊗ π Y ) ′ {\displaystyle \left(X\otimes _{\pi }Y\right)^{\prime }} is canonically identified with B ( X , Y ) {\displaystyle B(X,Y)} , the space of continuous bilinear maps on X × Y {\displaystyle X\times Y} . In this way, the continuous dual space of X ⊗ ϵ Y {\displaystyle X\otimes _{\epsilon }Y} can be canonically identified as a vector subspace of B ( X , Y ) {\displaystyle B(X,Y)} , denoted by J ( X , Y ) {\displaystyle J(X,Y)} . The elements of J ( X , Y ) {\displaystyle J(X,Y)} are called integral (bilinear) forms on X × Y {\displaystyle X\times Y} . The following theorem justifies the word integral.

Theorem¹²—The dual J(X, Y) of X ⊗ ^ ϵ Y {\displaystyle X{\widehat {\otimes }}_{\epsilon }Y} consists of exactly of the continuous bilinear forms u on X × Y {\displaystyle X\times Y} of the form

u ( x , y ) = ∫ S × T ⟨ x , x ′ ⟩ ⟨ y , y ′ ⟩ d μ ( x ′ , y ′ ) , {\displaystyle u(x,y)=\int _{S\times T}\langle x,x'\rangle \langle y,y'\rangle \;d\mu \!\left(x',y'\right),}

where S and T are respectively some weakly closed and equicontinuous (hence weakly compact) subsets of the duals X ′ {\displaystyle X^{\prime }} and Y ′ {\displaystyle Y^{\prime }} , and μ {\displaystyle \mu } is a (necessarily bounded) positive Radon measure on the (compact) set S × T {\displaystyle S\times T} .

There is also a closely related formulation ³ of the theorem above that can also be used to explain the terminology integral bilinear form: a continuous bilinear form u {\displaystyle u} on the product X × Y {\displaystyle X\times Y} of locally convex spaces is integral if and only if there is a compact topological space Ω {\displaystyle \Omega } equipped with a (necessarily bounded) positive Radon measure μ {\displaystyle \mu } and continuous linear maps α {\displaystyle \alpha } and β {\displaystyle \beta } from X {\displaystyle X} and Y {\displaystyle Y} to the Banach space L ∞ ( Ω , μ ) {\displaystyle L^{\infty }(\Omega ,\mu )} such that

u ( x , y ) = ⟨ α ( x ) , β ( y ) ⟩ = ∫ Ω α ( x ) β ( y ) d μ {\displaystyle u(x,y)=\langle \alpha (x),\beta (y)\rangle =\int _{\Omega }\alpha (x)\beta (y)\;d\mu } ,

i.e., the form u {\displaystyle u} can be realised by integrating (essentially bounded) functions on a compact space.

Integral linear maps

A continuous linear map κ : X → Y ′ {\displaystyle \kappa :X\to Y'} is called integral if its associated bilinear form is an integral bilinear form, where this form is defined by ( x , y ) ∈ X × Y ↦ ( κ x ) ( y ) {\displaystyle (x,y)\in X\times Y\mapsto (\kappa x)(y)} .⁴ It follows that an integral map κ : X → Y ′ {\displaystyle \kappa :X\to Y'} is of the form:⁵

x ∈ X ↦ κ ( x ) = ∫ S × T ⟨ x ′ , x ⟩ y ′ d μ ( x ′ , y ′ ) {\displaystyle x\in X\mapsto \kappa (x)=\int _{S\times T}\left\langle x',x\right\rangle y'\mathrm {d} \mu \!\left(x',y'\right)}

for suitable weakly closed and equicontinuous subsets S and T of X ′ {\displaystyle X'} and Y ′ {\displaystyle Y'} , respectively, and some positive Radon measure μ {\displaystyle \mu } of total mass ≤ 1. The above integral is the weak integral, so the equality holds if and only if for every y ∈ Y {\displaystyle y\in Y} , ⟨ κ ( x ) , y ⟩ = ∫ S × T ⟨ x ′ , x ⟩ ⟨ y ′ , y ⟩ d μ ( x ′ , y ′ ) {\textstyle \left\langle \kappa (x),y\right\rangle =\int _{S\times T}\left\langle x',x\right\rangle \left\langle y',y\right\rangle \mathrm {d} \mu \!\left(x',y'\right)} .

Given a linear map Λ : X → Y {\displaystyle \Lambda :X\to Y} , one can define a canonical bilinear form B Λ ∈ B i ( X , Y ′ ) {\displaystyle B_{\Lambda }\in Bi\left(X,Y'\right)} , called the associated bilinear form on X × Y ′ {\displaystyle X\times Y'} , by B Λ ( x , y ′ ) := ( y ′ ∘ Λ ) ( x ) {\displaystyle B_{\Lambda }\left(x,y'\right):=\left(y'\circ \Lambda \right)\left(x\right)} . A continuous map Λ : X → Y {\displaystyle \Lambda :X\to Y} is called integral if its associated bilinear form is an integral bilinear form.⁶ An integral map Λ : X → Y {\displaystyle \Lambda :X\to Y} is of the form, for every x ∈ X {\displaystyle x\in X} and y ′ ∈ Y ′ {\displaystyle y'\in Y'} :

⟨ y ′ , Λ ( x ) ⟩ = ∫ A ′ × B ″ ⟨ x ′ , x ⟩ ⟨ y ″ , y ′ ⟩ d μ ( x ′ , y ″ ) {\displaystyle \left\langle y',\Lambda (x)\right\rangle =\int _{A'\times B''}\left\langle x',x\right\rangle \left\langle y'',y'\right\rangle \mathrm {d} \mu \!\left(x',y''\right)}

for suitable weakly closed and equicontinuous aubsets A ′ {\displaystyle A'} and B ″ {\displaystyle B''} of X ′ {\displaystyle X'} and Y ″ {\displaystyle Y''} , respectively, and some positive Radon measure μ {\displaystyle \mu } of total mass ≤ 1 {\displaystyle \leq 1} .

Relation to Hilbert spaces

The following result shows that integral maps "factor through" Hilbert spaces.

Proposition:⁷ Suppose that u : X → Y {\displaystyle u:X\to Y} is an integral map between locally convex TVS with Y Hausdorff and complete. There exists a Hilbert space H and two continuous linear mappings α : X → H {\displaystyle \alpha :X\to H} and β : H → Y {\displaystyle \beta :H\to Y} such that u = β ∘ α {\displaystyle u=\beta \circ \alpha } .

Furthermore, every integral operator between two Hilbert spaces is nuclear.⁸ Thus a continuous linear operator between two Hilbert spaces is nuclear if and only if it is integral.

Sufficient conditions

Every nuclear map is integral.⁹ An important partial converse is that every integral operator between two Hilbert spaces is nuclear.¹⁰

Suppose that A, B, C, and D are Hausdorff locally convex TVSs and that α : A → B {\displaystyle \alpha :A\to B} , β : B → C {\displaystyle \beta :B\to C} , and γ : C → D {\displaystyle \gamma :C\to D} are all continuous linear operators. If β : B → C {\displaystyle \beta :B\to C} is an integral operator then so is the composition γ ∘ β ∘ α : A → D {\displaystyle \gamma \circ \beta \circ \alpha :A\to D} .¹¹

If u : X → Y {\displaystyle u:X\to Y} is a continuous linear operator between two normed space then u : X → Y {\displaystyle u:X\to Y} is integral if and only if t u : Y ′ → X ′ {\displaystyle {}^{t}u:Y'\to X'} is integral.¹²

Suppose that u : X → Y {\displaystyle u:X\to Y} is a continuous linear map between locally convex TVSs. If u : X → Y {\displaystyle u:X\to Y} is integral then so is its transpose t u : Y b ′ → X b ′ {\displaystyle {}^{t}u:Y_{b}^{\prime }\to X_{b}^{\prime }} .¹³ Now suppose that the transpose t u : Y b ′ → X b ′ {\displaystyle {}^{t}u:Y_{b}^{\prime }\to X_{b}^{\prime }} of the continuous linear map u : X → Y {\displaystyle u:X\to Y} is integral. Then u : X → Y {\displaystyle u:X\to Y} is integral if the canonical injections In X : X → X ″ {\displaystyle \operatorname {In} _{X}:X\to X''} (defined by x ↦ {\displaystyle x\mapsto } value at x) and In Y : Y → Y ″ {\displaystyle \operatorname {In} _{Y}:Y\to Y''} are TVS-embeddings (which happens if, for instance, X {\displaystyle X} and Y b ′ {\displaystyle Y_{b}^{\prime }} are barreled or metrizable).¹⁴

Properties

Suppose that A, B, C, and D are Hausdorff locally convex TVSs with B and D complete. If α : A → B {\displaystyle \alpha :A\to B} , β : B → C {\displaystyle \beta :B\to C} , and γ : C → D {\displaystyle \gamma :C\to D} are all integral linear maps then their composition γ ∘ β ∘ α : A → D {\displaystyle \gamma \circ \beta \circ \alpha :A\to D} is nuclear.¹⁵ Thus, in particular, if X is an infinite-dimensional Fréchet space then a continuous linear surjection u : X → X {\displaystyle u:X\to X} cannot be an integral operator.

See also

• Auxiliary normed spaces
• Final topology
• Injective tensor product
• Nuclear operators
• Nuclear spaces
• Projective tensor product
• Topological tensor product

Bibliography

• Diestel, Joe (2008). The Metric Theory of Tensor Products: Grothendieck's Résumé Revisited. Vol. 16. Providence, R.I.: American Mathematical Society. ISBN 9781470424831. OCLC 185095773.
• Dubinsky, Ed (1979). The Structure of Nuclear Fréchet Spaces. Lecture Notes in Mathematics. Vol. 720. Berlin New York: Springer-Verlag. ISBN 978-3-540-09504-0. OCLC 5126156.
• 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.
• Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Hogbe-Nlend, Henri (1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. Vol. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. MR 0500064. OCLC 316549583.
• Hogbe-Nlend, Henri; Moscatelli, V. B. (1981). Nuclear and Conuclear Spaces: Introductory Course on Nuclear and Conuclear Spaces in the Light of the Duality "topology-bornology". North-Holland Mathematics Studies. Vol. 52. Amsterdam New York New York: North Holland. ISBN 978-0-08-087163-9. OCLC 316564345.
• Pietsch, Albrecht (1979). Nuclear Locally Convex Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 66 (Second ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-05644-9. OCLC 539541.
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Ryan, Raymond A. (2002). Introduction to Tensor Products of Banach Spaces. Springer Monographs in Mathematics. London New York: Springer. ISBN 978-1-85233-437-6. 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.
• Wong, Yau-Chuen (1979). Schwartz Spaces, Nuclear Spaces, and Tensor Products. Lecture Notes in Mathematics. Vol. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.

External links

• Nuclear space at ncatlab

References

1. Schaefer & Wolff 1999, p. 168. - 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 ↩

2. Trèves 2006, pp. 500–502. - 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. Grothendieck 1955, pp. 124–126. - 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. https://mathscinet.ams.org/mathscinet-getitem?mr=0075539 ↩

4. Schaefer & Wolff 1999, p. 169. - 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 ↩

5. Schaefer & Wolff 1999, p. 169. - 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 ↩

6. Trèves 2006, pp. 502–505. - 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, pp. 506–508. - 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. Trèves 2006, pp. 506–508. - 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 ↩

9. Trèves 2006, pp. 502–505. - 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 ↩

10. Trèves 2006, pp. 506–508. - 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 ↩

11. Trèves 2006, pp. 506–508. - 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 ↩

12. Trèves 2006, pp. 505. - 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 ↩

13. Trèves 2006, pp. 502–505. - 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 ↩

14. Trèves 2006, pp. 502–505. - 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 ↩

15. Trèves 2006, pp. 506–508. - 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 ↩