and The kernel of is the subspace of orthogonal to the image of : 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 ). 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). (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 Duals of quotient spaces 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. 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 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. 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 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 The identity that characterizes the transpose, that is, is formally similar to the