Transpose of a linear map

Let X^# denote the algebraic dual space of a vector space X. Let X and Y be vector spaces over the same field 𝕂. If u : X → Y is a linear map, then its algebraic adjoint or dual,[1] is the map ^#u : Y^# → X^# defined by f ↦ f ∘ u. The resulting functional u^#(f) ≝ f ∘ u is called the pullback of f by u.

The continuous dual space of a topological vector space (TVS) X is denoted by X′. If X and Y are TVSs then a linear map u : X → Y is weakly continuous if and only if u^#(Y′) ⊆ X′, in which case we let ^tu : Y′ → X′ denote the restriction of u^# to Y′. The map ^tu is called the transpose[2] of u. The following identity characterizes the transpose of u[3]

⟨^tu(f), x⟩ = ⟨f, u(x)⟩ for all f ∈ Y′ and x ∈ X

where ⟨•, •⟩ is the natural pairing (i.e. defined by ⟨z, h⟩ ≝ h(z)).

The assignment u ↦ ^tu produces an injective linear map between the space of linear operators from X to Y and the space of linear operators from Y^# to X^#. If X = Y then the space of linear maps is an algebra under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that ^t(uv) = ^tv ^tu. 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 𝕂 to itself. One can identify ^t(^tu) with f using the natural injection into the double dual.

• If u : X → Y and v : Y → Z are linear maps then ^t(v ∘ u) = ^tu ∘ ^tv.[4]
• If u : X → Y is a linear map, A ⊆ X, B ⊆ Y, and A° denotes the polar set of the set A then[4]
  • [u(A)]° = (^tu)⁻¹(A°), and
  • u(A) ⊆ B implies ^tu(B°) ⊆ A°
• if A and B are convex, weakly closed sets containing 0 then ^tu(B°) ⊆ A° implies u(A) ⊆ B.[5]
• If u : X → Y is a (surjective) vector space isomorphism then so is the transpose ^tu : Y′ → X′.
• The kernel of ^tu is the subspace of Y′ orthogonal to the image of u.[5]
• The linear map u is injective if and only if the its image is a weakly dense subset of Y (i.e. the image of u is dense in Y when Y is given the weak topology induced by ker ^tu).[5]

Suppose now that u : X → Y is a continuous linear operator between topological vector spaces X and Y with continuous dual spaces X′ and Y′, respectively. For any subset S of X, let S° denote the polar of S in X′.

• The transpose ^tu : Y′ → X′ is continuous when both X′ and Y′ 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).[6]
• If X and Y are locally convex then ker ^tu = (Im u)°.[7]
• If X and Y are normed spaces then the norm of ^tu is equal to the norm of u.[7]
• (Surjection of Fréchet spaces): If X and Y are Fréchet spaces then the continuous linear operator u : X → Y is surjective if and only if (1) the transpose ^tu : Y′ → X′ is injective, and (2) the image of the transpose of u is a weakly closed (i.e. weak-* closed) subset of X′.[8]

If the linear map u is represented by the matrix A with respect to two bases of X and Y, then ^tu is represented by the transpose matrix A^T with respect to the dual bases of Y′ and X′, hence the name. Alternatively, as u is represented by A acting to the right on column vectors, ^tu 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 ℝⁿ, which identifies the space of column vectors with the dual space of row vectors.

The identity that characterizes the transpose, that is, [u^∗(f), x] = [f, u(x)], 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 Y′ → X′ and is defined for linear maps between any vector spaces X and Y, without requiring any additional structure. The Hermitian adjoint maps Y → X 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 Y → X. 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 X and Y are Hilbert spaces and u : X → Y is a linear map then the transpose of u and the Hermitian adjoint of u, which we will denote respectively by ^tu and u^∗, are related. Denote by I : X → X^∗ and J : Y → Y^∗ the canonical antilinear isometries of the Hilbert spaces X and Y onto their duals. Then u^∗ is the following composition of maps:[9]

${\displaystyle Y{\overset {J}{\longrightarrow }}Y^{*}{\overset {{}^{\text{t}}u}{\longrightarrow }}X^{*}{\overset {I^{-1}}{\longrightarrow }}X}$

Suppose that X and Y are topological vector spaces and that u : X → Y is a linear map, then many of u's properties are reflected in u^∗.

• If A ⊆ X and B ⊆ Y are weakly closed, convex sets containing 0, then u^∗(B°) ⊆ A° implies u(A) ⊆ B.[4]
• The null space of ^tu is the subspace of Y^∗ orthogonal to the range u(X) of u.[4]
• ^tu is injective if and only if the range u(X) of u is weakly closed.[4]

• Adjoint functors
• Hermitian adjoint
• Dual space § Transpose of a linear map
• Transpose § Transpose of a linear map

• Halmos, Paul (1974), Finite-dimensional Vector Spaces, Springer, ISBN 0-387-90093-4
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 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.