Injective tensor product

In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS without any need to extend definitions (such as "differentiable at a point") from real/complex-valued functions to -valued functions.

Preliminaries and notation

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Throughout let   and   be topological vector spaces and   be a linear map.

  •   is a topological homomorphism or homomorphism, if it is linear, continuous, and   is an open map, where   has the subspace topology induced by  
    • If   is a subspace of   then both the quotient map   and the canonical injection   are homomorphisms. In particular, any linear map   can be canonically decomposed as follows:   where   defines a bijection.
  • The set of continuous linear maps   (resp. continuous bilinear maps  ) will be denoted by   (resp.  ) where if   is the scalar field then we may instead write   (resp.  ).
  • The set of separately continuous bilinear maps   (that is, continuous in each variable when the other variable is fixed) will be denoted by   where if   is the scalar field then we may instead write  
  • We will denote the continuous dual space of   by   and the algebraic dual space (which is the vector space of all linear functionals on   whether continuous or not) by  
    • To increase the clarity of the exposition, we use the common convention of writing elements of   with a prime following the symbol (for example,   denotes an element of   and not, say, a derivative and the variables   and   need not be related in any way).

Notation for topologies

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Definition

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Throughout let   and   be topological vector spaces with continuous dual spaces   and   Note that almost all results described are independent of whether these vector spaces are over   or   but to simplify the exposition we will assume that they are over the field  

Continuous bilinear maps as a tensor product

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Despite the fact that the tensor product   is a purely algebraic construct (its definition does not involve any topologies), the vector space   of continuous bilinear functionals is nevertheless always a tensor product of   and   (that is,  ) when   is defined in the manner now described.[3]

For every   let   denote the bilinear form on   defined by   This map   is always continuous[3] and so the assignment that sends   to the bilinear form   induces a canonical map   whose image   is contained in   In fact, every continuous bilinear form on   belongs to the span of this map's image (that is,  ). The following theorem may be used to verify that   together with the above map   is a tensor product of   and  

Theorem — Let   and   be vector spaces and let   be a bilinear map. Then   is a tensor product of   and   if and only if[4] the image of   spans all of   (that is,  ), and the vectors spaces   and   are  -linearly disjoint, which by definition[5] means that for all sequences of elements   and   of the same finite length   satisfying  

  1. if all   are linearly independent then all   are   and
  2. if all   are linearly independent then all   are  

Equivalently,[4]   and   are  -linearly disjoint if and only if for all linearly independent sequences   in   and all linearly independent sequences   in   the vectors   are linearly independent.

Topology

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Henceforth, all topological vector spaces considered will be assumed to be locally convex. If   is any locally convex topological vector space, then  [6] and for any equicontinuous subsets   and   and any neighborhood   in   define   where every set   is bounded in  [6] which is necessary and sufficient for the collection of all   to form a locally convex TVS topology on  [7] This topology is called the  -topology and whenever a vector spaces is endowed with the  -topology then this will be indicated by placing   as a subscript before the opening parenthesis. For example,   endowed with the  -topology will be denoted by   If   is Hausdorff then so is the  -topology.[6]

In the special case where   is the underlying scalar field,   is the tensor product   and so the topological vector space   is called the injective tensor product of   and   and it is denoted by   This TVS is not necessarily complete so its completion, denoted by   will be constructed. When all spaces are Hausdorff then   is complete if and only if both   and   are complete,[8] in which case the completion   of   is a vector subspace of   If   and   are normed spaces then so is   where   is a Banach space if and only if this is true of both   and  [9]

Equicontinuous sets

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One reason for converging on equicontinuous subsets (of all possibilities) is the following important fact:

A set of continuous linear functionals   on a TVS  [note 1] is equicontinuous if and only if it is contained in the polar of some neighborhood   of the origin in  ; that is,  

A TVS's topology is completely determined by the open neighborhoods of the origin. This fact together with the bipolar theorem means that via the operation of taking the polar of a subset, the collection of all equicontinuous subsets of   "encodes" all information about  's given topology. Specifically, distinct locally convex TVS topologies on   produce distinct collections of equicontinuous subsets and conversely, given any such collection of equicontinuous sets, the TVS's original topology can be recovered by taking the polar of every (equicontinuous) set in the collection. Thus through this identification, uniform convergence on the collection of equicontinuous subsets is essentially uniform convergence on the very topology of the TVS; this allows one to directly relate the injective topology with the given topologies of   and   Furthermore, the topology of a locally convex Hausdorff space   is identical to the topology of uniform convergence on the equicontinuous subsets of  [10]

For this reason, the article now lists some properties of equicontinuous sets that are relevant for dealing with the injective tensor product. Throughout   and   are any locally convex space and   is a collection of linear maps from   into  

  • If   is equicontinuous then the subspace topologies that   inherits from the following topologies on   are identical:[11]
    1. the topology of precompact convergence;
    2. the topology of compact convergence;
    3. the topology of pointwise convergence;
    4. the topology of pointwise convergence on a given dense subset of  
  • An equicontinuous set   is bounded in the topology of bounded convergence (that is, bounded in  ).[11] So in particular,   will also bounded in every TVS topology that is coarser than the topology of bounded convergence.
  • If   is a barrelled space and   is locally convex then for any subset   the following are equivalent:
    1.   is equicontinuous;
    2.   is bounded in the topology of pointwise convergence (that is, bounded in  );
    3.   is bounded in the topology of bounded convergence (that is, bounded in  ).

In particular, to show that a set   is equicontinuous it suffices to show that it is bounded in the topology of pointwise converge.[12]

  • If   is a Baire space then any subset   that is bounded in   is necessarily equicontinuous.[12]
  • If   is separable,   is metrizable, and   is a dense subset of   then the topology of pointwise convergence on   makes   metrizable so that in particular, the subspace topology that any equicontinuous subset   inherits from   is metrizable.[11]

For equicontinuous subsets of the continuous dual space   (where   is now the underlying scalar field of  ), the following hold:

  • The weak closure of an equicontinuous set of linear functionals on   is a compact subspace of  [11]
  • If   is separable then every weakly closed equicontinuous subset of   is a metrizable compact space when it is given the weak topology (that is, the subspace topology inherited from  ).[11]
  • If   is a normable space then a subset   is equicontinuous if and only if it is strongly bounded (that is, bounded in  ).[11]
  • If   is a barrelled space then for any subset   the following are equivalent:[12]
    1.   is equicontinuous;
    2.   is relatively compact in the weak dual topology;
    3.   is weakly bounded;
    4.   is strongly bounded.

We mention some additional important basic properties relevant to the injective tensor product:

  • Suppose that   is a bilinear map where   is a Fréchet space,   is metrizable, and   is locally convex. If   is separately continuous then it is continuous.[13]

Canonical identification of separately continuous bilinear maps with linear maps

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The set equality   always holds; that is, if   is a linear map, then   is continuous if and only if   is continuous, where here   has its original topology.[14]

There also exists a canonical vector space isomorphism[14]   To define it, for every separately continuous bilinear form   defined on   and every   let   be defined by   Because   is canonically vector space-isomorphic to   (via the canonical map   value at  ),   will be identified as an element of   which will be denoted by   This defines a map   given by   and so the canonical isomorphism is of course defined by  

When   is given the topology of uniform convergence on equicontinous subsets of   the canonical map becomes a TVS-isomorphism[14]   In particular,   can be canonically TVS-embedded into  ; furthermore the image in   of   under the canonical map   consists exactly of the space of continuous linear maps   whose image is finite dimensional.[9]

The inclusion   always holds. If   is normed then   is in fact a topological vector subspace of   And if in addition   is Banach then so is   (even if   is not complete).[9]

Properties

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The canonical map   is always continuous[15] and the ε-topology is always coarser than the π-topology,[16] which is in turn coarser than the inductive topology (the finest locally convex TVS topology making   separately continuous). The space   is Hausdorff if and only if both   and   are Hausdorff.[15]

If   and   are normed then   is normable in which case for all    [17]

Suppose that   and   are two linear maps between locally convex spaces. If both   and   are continuous then so is their tensor product  [18] Moreover:

  • If   and   are both TVS-embeddings then so is  [19]
  • If   (resp.  ) is a linear subspace of   (resp.  ) then   is canonically isomorphic to a linear subspace of   and   is canonically isomorphic to a linear subspace of  [20]
  • There are examples of   and   such that both   and   are surjective homomorphisms but   is not a homomorphism.[21]
  • If all four spaces are normed then  [17]

Relation to projective tensor product and nuclear spaces

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The projective topology or the  -topology is the finest locally convex topology on   that makes continuous the canonical map   defined by sending   to the bilinear form   When   is endowed with this topology then it will be denoted by   and called the projective tensor product of   and  

The following definition was used by Grothendieck to define nuclear spaces.[22]

Definition 0: Let   be a locally convex topological vector space. Then   is nuclear if for any locally convex space   the canonical vector space embedding   is an embedding of TVSs whose image is dense in the codomain.

Canonical identifications of bilinear and linear maps

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In this section we describe canonical identifications between spaces of bilinear and linear maps. These identifications will be used to define important subspaces and topologies (particularly those that relate to nuclear operators and nuclear spaces).

Dual spaces of the injective tensor product and its completion

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Suppose that   denotes the TVS-embedding of   into its completion and let   be its transpose, which is a vector space-isomorphism. This identifies the continuous dual space of   as being identical to the continuous dual space of  

The identity map   is continuous (by definition of the π-topology) so there exists a unique continuous linear extension   If   and   are Hilbert spaces then   is injective and the dual of   is canonically isometrically isomorphic to the vector space   of nuclear operators from   into   (with the trace norm).

Injective tensor product of Hilbert spaces

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There is a canonical map   that sends   to the linear map   defined by   where it may be shown that the definition of   does not depend on the particular choice of representation   of   The map   is continuous and when   is complete, it has a continuous extension  

When   and   are Hilbert spaces then   is a TVS-embedding and isometry (when the spaces are given their usual norms) whose range is the space of all compact linear operators from   into   (which is a closed vector subspace of   Hence   is identical to space of compact operators from   into   (note the prime on  ). The space of compact linear operators between any two Banach spaces (which includes Hilbert spaces)   and   is a closed subset of  [23]

Furthermore, the canonical map   is injective when   and   are Hilbert spaces. [23]

Integral forms and operators

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Integral bilinear forms

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Denote the identity map by   and let   denote its transpose, which is a continuous injection. Recall that   is canonically identified with   the space of continuous bilinear maps on   In this way, the continuous dual space of   can be canonically identified as a subvector space of   denoted by   The elements of   are called integral (bilinear) forms on   The following theorem justifies the word integral.

Theorem[24][25] — The dual   of   consists of exactly those continuous bilinear forms v on   that can be represented in the form of a map   where   and   are some closed, equicontinuous subsets of   and   respectively, and   is a positive Radon measure on the compact set   with total mass   Furthermore, if   is an equicontinuous subset of   then the elements   can be represented with   fixed and   running through a norm bounded subset of the space of Radon measures on  

Integral linear operators

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Given a linear map   one can define a canonical bilinear form   called the associated bilinear form on   by   A continuous map   is called integral if its associated bilinear form is an integral bilinear form.[26] An integral map   is of the form, for every   and     for suitable weakly closed and equicontinuous subsets   and   of   and   respectively, and some positive Radon measure   of total mass  

Canonical map into L(X; Y)

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There is a canonical map   that sends   to the linear map   defined by   where it may be shown that the definition of   does not depend on the particular choice of representation   of  

Examples

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Space of summable families

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Throughout this section we fix some arbitrary (possibly uncountable) set   a TVS   and we let   be the directed set of all finite subsets of   directed by inclusion  

Let   be a family of elements in a TVS   and for every finite subset   let   We call   summable in   if the limit   of the net   converges in   to some element (any such element is called its sum). The set of all such summable families is a vector subspace of   denoted by  

We now define a topology on   in a very natural way. This topology turns out to be the injective topology taken from   and transferred to   via a canonical vector space isomorphism (the obvious one). This is a common occurrence when studying the injective and projective tensor products of function/sequence spaces and TVSs: the "natural way" in which one would define (from scratch) a topology on such a tensor product is frequently equivalent to the injective or projective tensor product topology.

Let   denote a base of convex balanced neighborhoods of 0 in   and for each   let   denote its Minkowski functional. For any such   and any   let   where   defines a seminorm on   The family of seminorms   generates a topology making   into a locally convex space. The vector space   endowed with this topology will be denoted by  [27] The special case where   is the scalar field will be denoted by  

There is a canonical embedding of vector spaces   defined by linearizing the bilinear map   defined by  [27]

Theorem:[27] — The canonical embedding (of vector spaces)   becomes an embedding of topological vector spaces   when   is given the injective topology and furthermore, its range is dense in its codomain. If   is a completion of   then the continuous extension   of this embedding   is an isomorphism of TVSs. So in particular, if   is complete then   is canonically isomorphic to  

Space of continuously differentiable vector-valued functions

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Throughout, let   be an open subset of   where   is an integer and let   be a locally convex topological vector space (TVS).

Definition[28] Suppose   and   is a function such that   with   a limit point of   Say that   is differentiable at   if there exist   vectors   in   called the partial derivatives of  , such that   where  

One may naturally extend the notion of continuously differentiable function to  -valued functions defined on   For any   let   denote the vector space of all    -valued maps defined on   and let   denote the vector subspace of   consisting of all maps in   that have compact support.

One may then define topologies on   and   in the same manner as the topologies on   and   are defined for the space of distributions and test functions (see the article: Differentiable vector-valued functions from Euclidean space). All of this work in extending the definition of differentiability and various topologies turns out to be exactly equivalent to simply taking the completed injective tensor product:

Theorem[29] — If   is a complete Hausdorff locally convex space, then   is canonically isomorphic to the injective tensor product  

Spaces of continuous maps from a compact space

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If   is a normed space and if   is a compact set, then the  -norm on   is equal to  [29] If   and   are two compact spaces, then   where this canonical map is an isomorphism of Banach spaces.[29]

Spaces of sequences converging to 0

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If   is a normed space, then let   denote the space of all sequences   in   that converge to the origin and give this space the norm   Let   denote   Then for any Banach space     is canonically isometrically isomorphic to  [29]

Schwartz space of functions

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We will now generalize the Schwartz space to functions valued in a TVS. Let   be the space of all   such that for all pairs of polynomials   and   in   variables,   is a bounded subset of   To generalize the topology of the Schwartz space to   we give   the topology of uniform convergence over   of the functions   as   and   vary over all possible pairs of polynomials in   variables.[29]

Theorem[29] — If   is a complete locally convex space, then   is canonically isomorphic to  

See also

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Notes

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  1. ^ This is true even if   is not assumed to be Hausdorff or locally convex.

References

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  1. ^ Trèves 2006, pp. 432–434.
  2. ^ Trèves 2006, pp. 338–345.
  3. ^ a b Trèves 2006, pp. 431–432.
  4. ^ a b Trèves 2006, pp. 403–404.
  5. ^ Trèves 2006, p. 403.
  6. ^ a b c Trèves 2006, p. 428.
  7. ^ Trèves 2006, pp. 427–428.
  8. ^ Trèves 2006, p. 430.
  9. ^ a b c Trèves 2006, pp. 432–433.
  10. ^ Trèves 2006, pp. 368–370.
  11. ^ a b c d e f Trèves 2006, pp. 338–343.
  12. ^ a b c Trèves 2006, pp. 347–350.
  13. ^ Trèves 2006, pp. 351–354.
  14. ^ a b c Trèves 2006, pp. 428–430.
  15. ^ a b Trèves 2006, p. 434.
  16. ^ Trèves 2006, p. 438.
  17. ^ a b Trèves 2006, p. 444.
  18. ^ Trèves 2006, p. 439.
  19. ^ Trèves 2006, p. 440.
  20. ^ Trèves 2006, p. 441.
  21. ^ Trèves 2006, p. 442.
  22. ^ Schaefer & Wolff 1999, p. 170.
  23. ^ a b Trèves 2006, p. 494.
  24. ^ Schaefer & Wolff 1999, p. 168.
  25. ^ Trèves 2006, pp. 500–502.
  26. ^ Trèves 2006, pp. 502–505.
  27. ^ a b c Schaefer & Wolff 1999, pp. 179–184.
  28. ^ Trèves 2006, pp. 412–419.
  29. ^ a b c d e f Trèves 2006, pp. 446–451.

Bibliography

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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.
  • Dubinsky, Ed (1979). The structure of nuclear Fréchet spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09504-7. OCLC 5126156.
  • Grothendieck, Grothendieck (1966). Produits tensoriels topologiques et espaces nucléaires (in French). Providence: American Mathematical Society. ISBN 0-8218-1216-5. OCLC 1315788.
  • Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. 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.
  • Nlend, H (1977). Bornologies and functional analysis : introductory course on the theory of duality topology-bornology and its use in functional analysis. Amsterdam New York New York: North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier-North Holland. ISBN 0-7204-0712-5. OCLC 2798822.
  • Nlend, H (1981). Nuclear and conuclear spaces : introductory courses on nuclear and conuclear spaces in the light of the duality. Amsterdam New York New York, N.Y: North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier North-Holland. ISBN 0-444-86207-2. OCLC 7553061.
  • Pietsch, Albrecht (1972). Nuclear locally convex spaces. Berlin, New York: Springer-Verlag. ISBN 0-387-05644-0. OCLC 539541.
  • Robertson, A. P. (1973). Topological vector spaces. Cambridge England: University Press. ISBN 0-521-29882-2. OCLC 589250.
  • 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.
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