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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 27th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexhad occasion to create _[[dual vector space]]_.

   had occasion to create dual vector space.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJan 30th 2012
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI added > $V \mapsto V^*$ extends to a contravariant functor: given a linear map $A: V\to W$, the dual or __transpose map__ $A^* = A^T : W^*\to V^*$ is given by $\langle A^*(w_1), v\rangle = \langle w_1, A(v)\rangle$ $\forall w_1\in W^*$, $\forall v\in V$.

   I added

   $V \mapsto V^*$ extends to a contravariant functor: given a linear map $A: V\to W$, the dual or transpose map $A^* = A^T : W^*\to V^*$ is given by $\langle A^*(w_1), v\rangle = \langle w_1, A(v)\rangle$ $\forall w_1\in W^*$, $\forall v\in V$.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeSep 5th 2013
   • (edited Sep 5th 2013)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexNew related entry (only a small part of intended material covered so far) [[dual gebra]].

   New related entry (only a small part of intended material covered so far) dual gebra.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeFeb 4th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have touched the wording in the section "Double dual" ([here](https://ncatlab.org/nlab/show/dual+vector+space#DoubleDualVectorSpaces)), for streamlining. <a href="https://ncatlab.org/nlab/revision/diff/dual+vector+space/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/dual+vector+space/23">v23</a>, <a href="https://ncatlab.org/nlab/show/dual+vector+space">current</a>

   I have touched the wording in the section “Double dual” (here), for streamlining.

   diff, v23, current