Inductive tensor product

The finest locally convex topological vector space (TVS) topology on the tensor product of two locally convex TVSs, making the canonical map (defined by sending to ) separately continuous is called the inductive topology or the -topology. When is endowed with this topology then it is denoted by and called the inductive tensor product of and [1]

Preliminaries

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Throughout let   and   be locally convex 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   the image of   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.  ).
  • 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 (e.g.   denotes an element of   and not, say, a derivative and the variables   and   need not be related in any way).
  • A linear map   from a Hilbert space into itself is called positive if   for every   In this case, there is a unique positive map   called the square-root of   such that  [2]
    • If   is any continuous linear map between Hilbert spaces, then   is always positive. Now let   denote its positive square-root, which is called the absolute value of   Define   first on   by setting   for   and extending   continuously to   and then define   on   by setting   for   and extend this map linearly to all of   The map   is a surjective isometry and  
  • A linear map   is called compact or completely continuous if there is a neighborhood   of the origin in   such that   is precompact in  [3]
    • In a Hilbert space, positive compact linear operators, say   have a simple spectral decomposition discovered at the beginning of the 20th century by Fredholm and F. Riesz:[4]
There is a sequence of positive numbers, decreasing and either finite or else converging to 0,   and a sequence of nonzero finite dimensional subspaces   of   ( ) with the following properties: (1) the subspaces   are pairwise orthogonal; (2) for every   and every    ; and (3) the orthogonal of the subspace spanned by   is equal to the kernel of  [4]

Notation for topologies

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Universal property

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Suppose that   is a locally convex space and that   is the canonical map from the space of all bilinear mappings of the form   going into the space of all linear mappings of  [1] Then when the domain of   is restricted to   (the space of separately continuous bilinear maps) then the range of this restriction is the space   of continuous linear operators   In particular, the continuous dual space of   is canonically isomorphic to the space   the space of separately continuous bilinear forms on  

If   is a locally convex TVS topology on   (  with this topology will be denoted by  ), then   is equal to the inductive tensor product topology if and only if it has the following property:[5]

For every locally convex TVS   if   is the canonical map from the space of all bilinear mappings of the form   going into the space of all linear mappings of   then when the domain of   is restricted to   (space of separately continuous bilinear maps) then the range of this restriction is the space   of continuous linear operators  

See also

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References

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  1. ^ a b Schaefer & Wolff 1999, p. 96.
  2. ^ Trèves 2006, p. 488.
  3. ^ Trèves 2006, p. 483.
  4. ^ a b Trèves 2006, p. 490.
  5. ^ Grothendieck 1966, p. 73.

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, Alexander (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.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
  • 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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