Editing Transpose (section)

=== Transpose of a linear map ===
{{Main|Transpose of a linear map}}

Let {{math|''X''<sup>#</sup>}} denote the [[algebraic dual space]] of an {{mvar|R}}-[[Module (mathematics)|module]] {{mvar|X}}. 
Let {{mvar|X}} and {{mvar|Y}} be {{mvar|R}}-modules. 
If {{math|''u'' : ''X'' → ''Y''}} is a [[linear map]], then its '''algebraic adjoint''' or '''dual''',{{sfn | Schaefer | Wolff | 1999 | p=128}} is the map {{math|''u''<sup>#</sup> : ''Y''<sup>#</sup> &rarr; ''X''<sup>#</sup>}} defined by {{math|''f'' {{mapsto}} ''f'' ∘ ''u''}}. 
The resulting functional {{math|''u''<sup>#</sup>(''f'')}} is called the '''[[pullback (differential geometry)|pullback]]''' of {{mvar|f}} by {{mvar|u}}. 
The following [[Relation (math)|relation]] characterizes the algebraic adjoint of {{mvar|u}}<ref>{{harvnb|Halmos|1974|loc=§44}}</ref> 
:{{math|{{angbr|''u''<sup>#</sup>(''f''), ''x''}} {{=}} {{angbr|''f'', ''u''(''x'')}}}} for all {{math|''f'' ∈ ''Y''<sup>#</sup>}} and {{math|''x'' ∈ ''X''}}
where {{math|{{angbr|&bull;, &bull;}}}} is the [[natural pairing]] (i.e. defined by {{math|{{angbr|''h'', ''z''}} :{{=}} ''h''(''z'')}}). 
This definition also applies unchanged to left modules and to vector spaces.<ref>{{harvnb|Bourbaki|1989|loc=II §2.5 }}</ref>

The definition of the transpose may be seen to be independent of any bilinear form on the modules, unlike the adjoint ([[#Adjoint|below]]).

The [[continuous dual space]] of a [[topological vector space]] (TVS) {{mvar|X}} is denoted by {{math|''X''{{big|{{'}}}}}}. 
If {{mvar|X}} and {{mvar|Y}} are TVSs then a linear map {{math|''u'' : ''X'' → ''Y''}} is '''weakly continuous''' if and only if {{math|''u''<sup>#</sup>(''Y''{{big|{{'}}}}) &sube; ''X''{{big|{{'}}}}}}, in which case we let {{math|<sup>t</sup>''u'' : ''Y''{{big|{{'}}}} &rarr; ''X''{{big|{{'}}}}}} denote the restriction of {{math|''u''<sup>#</sup>}} to {{math|''Y''{{big|{{'}}}}}}. 
The map {{math|<sup>t</sup>''u''}} is called the '''transpose'''{{sfn | Trèves | 2006 | p=240}} of {{mvar|u}}. 

If the matrix {{math|'''A'''}} describes a linear map with respect to [[basis (linear algebra)|bases]] of {{mvar|V}} and {{mvar|W}}, then the matrix {{math|'''A'''<sup>T</sup>}} describes the transpose of that linear map with respect to the [[Dual basis|dual bases]].