Transpose of a linear map


In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose or algebraic adjoint of a linear map is often used to study the original linear map. This concept is generalised by adjoint functors.

Definition edit

Let   denote the algebraic dual space of a vector space   Let   and   be vector spaces over the same field   If   is a linear map, then its algebraic adjoint or dual,[1] is the map   defined by   The resulting functional   is called the pullback of   by  

The continuous dual space of a topological vector space (TVS)   is denoted by   If   and   are TVSs then a linear map   is weakly continuous if and only if   in which case we let   denote the restriction of   to   The map   is called the transpose[2] or algebraic adjoint of   The following identity characterizes the transpose of  :[3]

where   is the natural pairing defined by  

Properties edit

The assignment   produces an injective linear map between the space of linear operators from   to   and the space of linear operators from   to   If   then the space of linear maps is an algebra under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that   In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over   to itself. One can identify   with   using the natural injection into the double dual.

  • If   and   are linear maps then  [4]
  • If   is a (surjective) vector space isomorphism then so is the transpose  
  • If   and   are normed spaces then

and if the linear operator   is bounded then the operator norm of   is equal to the norm of  ; that is[5][6]
and moreover,

Polars edit

Suppose now that   is a weakly continuous linear operator between topological vector spaces   and   with continuous dual spaces   and   respectively. Let   denote the canonical dual system, defined by   where   and   are said to be orthogonal if   For any subsets   and   let

denote the (absolute) polar of   in   (resp. of   in  ).
  • If   and   are convex, weakly closed sets containing the origin then   implies  [7]
  • If   and   then[4]



Annihilators edit

Suppose   and   are topological vector spaces and   is a weakly continuous linear operator (so  ). Given subsets   and   define their annihilators (with respect to the canonical dual system) by[6]



  • The kernel of   is the subspace of   orthogonal to the image of  :[7]

  • The linear map   is injective if and only if its image is a weakly dense subset of   (that is, the image of   is dense in   when   is given the weak topology induced by  ).[7]
  • The transpose   is continuous when both   and   are endowed with the weak-* topology (resp. both endowed with the strong dual topology, both endowed with the topology of uniform convergence on compact convex subsets, both endowed with the topology of uniform convergence on compact subsets).[8]
  • (Surjection of Fréchet spaces): If   and   are Fréchet spaces then the continuous linear operator   is surjective if and only if (1) the transpose   is injective, and (2) the image of the transpose of   is a weakly closed (i.e. weak-* closed) subset of  [9]

Duals of quotient spaces edit

Let   be a closed vector subspace of a Hausdorff locally convex space   and denote the canonical quotient map by

Assume   is endowed with the quotient topology induced by the quotient map   Then the transpose of the quotient map is valued in   and
is a TVS-isomorphism onto   If   is a Banach space then   is also an isometry.[6] Using this transpose, every continuous linear functional on the quotient space   is canonically identified with a continuous linear functional in the annihilator   of  

Duals of vector subspaces edit

Let   be a closed vector subspace of a Hausdorff locally convex space   If   and if   is a continuous linear extension of   to   then the assignment   induces a vector space isomorphism

which is an isometry if   is a Banach space.[6]

Denote the inclusion map by

The transpose of the inclusion map is
whose kernel is the annihilator   and which is surjective by the Hahn–Banach theorem. This map induces an isomorphism of vector spaces

Representation as a matrix edit

If the linear map   is represented by the matrix   with respect to two bases of   and   then   is represented by the transpose matrix   with respect to the dual bases of   and   hence the name. Alternatively, as   is represented by   acting to the right on column vectors,   is represented by the same matrix acting to the left on row vectors. These points of view are related by the canonical inner product on   which identifies the space of column vectors with the dual space of row vectors.

Relation to the Hermitian adjoint edit

The identity that characterizes the transpose, that is,   is formally similar to the definition of the Hermitian adjoint, however, the transpose and the Hermitian adjoint are not the same map. The transpose is a map   and is defined for linear maps between any vector spaces   and   without requiring any additional structure. The Hermitian adjoint maps   and is only defined for linear maps between Hilbert spaces, as it is defined in terms of the inner product on the Hilbert space. The Hermitian adjoint therefore requires more mathematical structure than the transpose.

However, the transpose is often used in contexts where the vector spaces are both equipped with a nondegenerate bilinear form such as the Euclidean dot product or another real inner product. In this case, the nondegenerate bilinear form is often used implicitly to map between the vector spaces and their duals, to express the transposed map as a map   For a complex Hilbert space, the inner product is sesquilinear and not bilinear, and these conversions change the transpose into the adjoint map.

More precisely: if   and   are Hilbert spaces and   is a linear map then the transpose of   and the Hermitian adjoint of   which we will denote respectively by   and   are related. Denote by   and   the canonical antilinear isometries of the Hilbert spaces   and   onto their duals. Then   is the following composition of maps:[10]


Applications to functional analysis edit

Suppose that   and   are topological vector spaces and that   is a linear map, then many of  's properties are reflected in  

  • If   and   are weakly closed, convex sets containing the origin, then   implies  [4]
  • The null space of   is the subspace of   orthogonal to the range   of  [4]
  •   is injective if and only if the range   of   is weakly closed.[4]

See also edit

References edit

  1. ^ Schaefer & Wolff 1999, p. 128.
  2. ^ Trèves 2006, p. 240.
  3. ^ Halmos (1974, §44)
  4. ^ a b c d e Schaefer & Wolff 1999, pp. 129–130
  5. ^ a b Trèves 2006, pp. 240–252.
  6. ^ a b c d Rudin 1991, pp. 92–115.
  7. ^ a b c Schaefer & Wolff 1999, pp. 128–130.
  8. ^ Trèves 2006, pp. 199–200.
  9. ^ Trèves 2006, pp. 382–383.
  10. ^ Trèves 2006, p. 488.

Bibliography edit

  • Halmos, Paul (1974), Finite-dimensional Vector Spaces, Springer, ISBN 0-387-90093-4
  • 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.
  • 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.