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. Dual system#Transposes and Transpose#Transposes of linear maps and bilinear forms 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, 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 or algebraic adjoint of The following identity characterizes the transpose of where is the natural pairing defined by 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 , 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 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 and moreover 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 if For any subsets and let denote the () (resp. ).