In mathematical analysis, anintegral linear operator is a linear operatorT given by integration; i.e.,

${\displaystyle (Tf)(x)=\int f(y)K(x,y)dy}$

where${\displaystyle K(x,y)}$ is called an integration kernel.

More generally, anintegral bilinear form is abilinear functional that belongs to the continuous dual space of${\displaystyle X{\hat{\otimes }}_{ϵ}Y}$, theinjective tensor product of the locally convextopological vector spaces (TVSs)X andY. Anintegral linear operator is a continuous linear operator that arises in a canonical way from an integral bilinear form.

These maps play an important role in the theory ofnuclear spaces andnuclear maps.

LetX andY be locally convex TVSs, let${\displaystyle X{\otimes }_{\pi }Y}$  denote theprojective tensor product,${\displaystyle X{\hat{\otimes }}_{\pi }Y}$  denote its completion, let${\displaystyle X{\otimes }_{ϵ}Y}$  denote theinjective tensor product, and${\displaystyle X{\hat{\otimes }}_{ϵ}Y}$  denote its completion. Suppose that${\displaystyle \mathrm{In}:X{\otimes }_{ϵ}Y\to X{\hat{\otimes }}_{ϵ}Y}$  denotes the TVS-embedding of${\displaystyle X{\otimes }_{ϵ}Y}$  into its completion and let${\displaystyle {}^t\mathrm{In}:{\left(X{\hat{\otimes }}_{ϵ}Y\right)}_b^{\mathrm{\prime }}\to {\left(X{\otimes }_{ϵ}Y\right)}_b^{\mathrm{\prime }}}$  be itstranspose, which is a vector space-isomorphism. This identifies the continuous dual space of${\displaystyle X{\otimes }_{ϵ}Y}$  as being identical to the continuous dual space of${\displaystyle X{\hat{\otimes }}_{ϵ}Y}$ .

Let${\displaystyle \mathrm{Id}:X{\otimes }_{\pi }Y\to X{\otimes }_{ϵ}Y}$  denote the identity map and${\displaystyle {}^t\mathrm{Id}:{\left(X{\otimes }_{ϵ}Y\right)}_b^{\mathrm{\prime }}\to {\left(X{\otimes }_{\pi }Y\right)}_b^{\mathrm{\prime }}}$  denote itstranspose, which is a continuous injection. Recall that${\displaystyle {\left(X{\otimes }_{\pi }Y\right)}^{\mathrm{\prime }}}$  is canonically identified with${\displaystyle B(X,Y)}$ , the space of continuous bilinear maps on${\displaystyle X\times Y}$ . In this way, the continuous dual space of${\displaystyle X{\otimes }_{ϵ}Y}$  can be canonically identified as a vector subspace of${\displaystyle B(X,Y)}$ , denoted by${\displaystyle J(X,Y)}$ . The elements of${\displaystyle J(X,Y)}$  are calledintegral (bilinear) forms on${\displaystyle X\times Y}$ . The following theorem justifies the wordintegral.

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

${\displaystyle u(x,y)=⟨\alpha (x),\beta (y)⟩={\int }_{\mathrm{\Omega }}\alpha (x)\beta (y)d\mu }$ ,

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

A continuous linear map${\displaystyle \kappa :X\to Y^{\prime }}$  is calledintegral if its associated bilinear form is an integral bilinear form, where this form is defined by${\displaystyle (x,y)\in X\times Y\mapsto (\kappa x)(y)}$ .^[4] It follows that an integral map${\displaystyle \kappa :X\to Y^{\prime }}$  is of the form:^[4]

${\displaystyle x\in X\mapsto \kappa (x)={\int }_{S\times T}⟨x^{\prime },x⟩y^{\prime }\mathrm{d}\mu \left(x^{\prime },y^{\prime }\right)}$ 

for suitable weakly closed and equicontinuous subsetsS andT of${\displaystyle X^{\prime }}$  and${\displaystyle Y^{\prime }}$ , respectively, and some positive Radon measure${\displaystyle \mu }$  of total mass ≤ 1. The above integral is theweak integral, so the equality holds if and only if for every${\displaystyle y\in Y}$ ,$⟨\kappa (x),y⟩={\int }_{S\times T}⟨x^{\prime },x⟩⟨y^{\prime },y⟩\mathrm{d}\mu \left(x^{\prime },y^{\prime }\right)$ .

Given a linear map${\displaystyle \mathrm{\Lambda }:X\to Y}$ , one can define a canonical bilinear form${\displaystyle B_{\mathrm{\Lambda }}\in Bi\left(X,Y^{\prime }\right)}$ , called theassociated bilinear form on${\displaystyle X\times Y^{\prime }}$ , by${\displaystyle B_{\mathrm{\Lambda }}\left(x,y^{\prime }\right):=\left(y^{\prime }\circ \mathrm{\Lambda }\right)\left(x\right)}$ . A continuous map${\displaystyle \mathrm{\Lambda }:X\to Y}$  is calledintegral if its associated bilinear form is an integral bilinear form.^[5] An integral map${\displaystyle \mathrm{\Lambda }:X\to Y}$  is of the form, for every${\displaystyle x\in X}$  and${\displaystyle y^{\prime }\in Y^{\prime }}$ :

${\displaystyle ⟨y^{\prime },\mathrm{\Lambda }(x)⟩={\int }_{A^{\prime }\times B^{″}}⟨x^{\prime },x⟩⟨y^{″},y^{\prime }⟩\mathrm{d}\mu \left(x^{\prime },y^{″}\right)}$ 

for suitable weakly closed and equicontinuous aubsets${\displaystyle A^{\prime }}$  and${\displaystyle B^{″}}$  of${\displaystyle X^{\prime }}$  and${\displaystyle Y^{″}}$ , respectively, and some positive Radon measure${\displaystyle \mu }$  of total mass${\displaystyle \le 1}$ .

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

Proposition:^[6] Suppose that${\displaystyle u:X\to Y}$  is an integral map between locally convex TVS withY Hausdorff and complete. There exists a Hilbert spaceH and two continuous linear mappings${\displaystyle \alpha :X\to H}$  and${\displaystyle \beta :H\to Y}$  such that${\displaystyle u=\beta \circ \alpha }$ .

Furthermore, every integral operator between twoHilbert spaces isnuclear.^[6] Thus a continuous linear operator between twoHilbert spaces isnuclear if and only if it is integral.

Everynuclear map is integral.^[5] An important partial converse is that every integral operator between twoHilbert spaces isnuclear.^[6]

Suppose thatA,B,C, andD are Hausdorff locally convex TVSs and that${\displaystyle \alpha :A\to B}$ ,${\displaystyle \beta :B\to C}$ , and${\displaystyle \gamma :C\to D}$  are all continuous linear operators. If${\displaystyle \beta :B\to C}$  is an integral operator then so is the composition${\displaystyle \gamma \circ \beta \circ \alpha :A\to D}$ .^[6]

If${\displaystyle u:X\to Y}$  is a continuous linear operator between two normed space then${\displaystyle u:X\to Y}$  is integral if and only if${\displaystyle {}^{t}u:Y^{\prime }\to X^{\prime }}$  is integral.^[7]

Suppose that${\displaystyle u:X\to Y}$  is a continuous linear map between locally convex TVSs. If${\displaystyle u:X\to Y}$  is integral then so is itstranspose${\displaystyle {}^{t}u:Y_b^{\mathrm{\prime }}\to X_b^{\mathrm{\prime }}}$ .^[5] Now suppose that the transpose${\displaystyle {}^{t}u:Y_b^{\mathrm{\prime }}\to X_b^{\mathrm{\prime }}}$  of the continuous linear map${\displaystyle u:X\to Y}$  is integral. Then${\displaystyle u:X\to Y}$  is integral if the canonical injections${\displaystyle {\mathrm{In}}_X:X\to X^{″}}$  (defined by${\displaystyle x\mapsto }$  value atx) and${\displaystyle {\mathrm{In}}_Y:Y\to Y^{″}}$  areTVS-embeddings (which happens if, for instance,${\displaystyle X}$  and${\displaystyle Y_b^{\mathrm{\prime }}}$  are barreled or metrizable).^[5]

Suppose thatA,B,C, andD are Hausdorff locally convex TVSs withB andDcomplete. If${\displaystyle \alpha :A\to B}$ ,${\displaystyle \beta :B\to C}$ , and${\displaystyle \gamma :C\to D}$  are all integral linear maps then their composition${\displaystyle \gamma \circ \beta \circ \alpha :A\to D}$  isnuclear.^[6] Thus, in particular, ifX is an infinite-dimensionalFréchet space then a continuous linear surjection${\displaystyle u:X\to X}$  cannot be an integral operator.

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

• 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.MR 0075539.OCLC 9308061.
• 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.

• Nuclear space at ncatlab