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\begin{document}

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\section*{dual vector space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{duality}{}\paragraph*{{Duality}}\label{duality}

[[!include duality - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{transpose_maps}{Transpose maps}\dotfill \pageref*{transpose_maps} \linebreak
\noindent\hyperlink{dual_bases}{Dual bases}\dotfill \pageref*{dual_bases} \linebreak
\noindent\hyperlink{double_duals}{Double duals}\dotfill \pageref*{double_duals} \linebreak
\noindent\hyperlink{dual_spaces_as_dual_objects}{Dual spaces as dual objects}\dotfill \pageref*{dual_spaces_as_dual_objects} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{dual vector space} is a [[dual in a closed category]] of [[vector spaces]] (or similar [[algebraic structures]]).

Of course, this is a very restricted notion of [[space]]; but for spaces in [[geometry]], one usually uses the [[duality between space and quantity]] and speaks of the [[spectrum]] (not `dual space') of an [[algebra]]. In [[homotopy theory]], there are also [[Spanier-Whitehead dual]]s; and there are other notions of dual space in [[physics]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Let $K$ be a [[field]] (or any [[commutative ring|commutative]] [[rig]]), and let $V$ be a [[vector space]] (or [[module]]) over $K$.

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{dual space} or \textbf{dual module} of $V$ is the vector space $V^*$ of [[linear functionals]] on $V$. That is, $V^*$ is the [[internal hom]] $[V,K]$ (thinking of $K$ as a vector space over itself: a [[line]]).

\end{defn}
More generally, let $K$ and $L$ be [[rings]] (or [[rigs]]) \emph{not} assumed commutative, and let $V$ be a $K$-$L$-[[bimodule]].

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{left dual module} of $V$ is the right $K$-module $^*V$ of left $K$-module [[homomorphisms]] from $V$ to $K$. The \textbf{right dual module} of $V$ is the left $L$-module $V^*$ of right $L$-module homomorphisms from $V$ to $L$.

\end{defn}
Now let $V$ be a [[topological vector space]] over the [[ground field]] $K$.

\begin{defn}
\label{LinearDualOfATopologicalVectorSpace}\hypertarget{LinearDualOfATopologicalVectorSpace}{}
\textbf{(linear dual of a topological vector space)}

Let $V$ be a [[topological vector space]] over the [[ground field]] $K$.

The \textbf{dual space} of $V$ is the topological vector space $V^*$ of [[continuous linear functionals]] on $V$, equipped with the [[weak-star topology|weak-* topology]] (meaning the [[initial topology]] generated by the elements of $V$, viewed as themselves linear functionals on $V^*$).

\end{defn}
In principle, there is no conflict among these definitions, the most general case (so far) being a topological bimodule over two topological rigs; the non-topological cases simply involve [[discrete spaces]]. In practice, however, some complications are possible:

\begin{itemize}%
\item If the rig $K$ is an [[associative algebra|algebra]] over another rig $L$, then any $K$-module $V$ is also an $L$-module, but the dual as a $K$-module is different from the dual as an $L$-module. So one may speak of the \textbf{$K$-dual} or the \textbf{dual over $K$}.


\item A topological vector space $V$ has an [[underlying]] discrete vector space, and these have different duals. So one speaks of the \textbf{topological dual} and the \textbf{algebraic dual} (respectively). If $V$ is considered with several different topologies (say called `weak' and `strong'), then one may speak of the \textbf{weak dual} and the \textbf{strong dual} (etc).



\end{itemize}
Logically, these duals take place in different [[categories]], which are related by various [[functors]]; the objects whose duals are being taken should not be conflated. In practice, however, these objects are identified, so the duals must be distinguished.

\hypertarget{transpose_maps}{}\subsection*{{Transpose maps}}\label{transpose_maps}

The operation $V \mapsto V^*$ extends to a [[contravariant functor]].

\begin{defn}
\label{}\hypertarget{}{}
The [[dual linear map]] or \textbf{transpose map} of a [[linear map]] $A\colon V\to W$, is the linear map $A^* = A^T\colon W^*\to V^*$, given by

\begin{displaymath}
\langle{A^*(w), v}\rangle = \langle{w, A(v)}\rangle
\end{displaymath}
for all $w$ in $W^*$ and $v$ in $V$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This functor is, of course, the [[representable functor]] represented by $K$ as a vector space over itself (a [[line]]).

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
This construction is the notion of [[dual morphism]] applied in the [[monoidal category]] [[Vect]] with its [[tensor product]] monoidal structure.

\end{remark}
\hypertarget{dual_bases}{}\subsection*{{Dual bases}}\label{dual_bases}

If $B$ is any [[basis]] of $V$, then we can sometimes turn $B$ into a basis $B^*$ of the dual space $V^*$.

We will begin with the definition of what \emph{might} be the dual basis, cautioning that it is not always a basis:

\begin{defn}
\label{}\hypertarget{}{}
Treating the basis $B$ as a [[family]] $(b_i)$ with index set $I$, the \textbf{dual basis} $B^*$ is the family $(b^i)$ (with the same index set) such that

\begin{displaymath}
b^i(b_j) = \delta^i_j
\end{displaymath}
(the [[Kronecker delta]]).

\end{defn}
Since $B$ is a basis of $V$, this formula defines $b^i$ for each index $I$, so $B^*$ exists; but in general there is no reason why $B^*$ should be a basis of $V^*$. However, if $V$ has [[finite set|finite]] [[dimension]], then $B^*$ is a basis of $V^*$. If $V$ is a [[Hilbert space]], and $B$ is a basis of $V$ in the Hilbert space sense (i.e., $B$ is a linearly independent set whose span is topologically dense in $V$), then also $B^*$ is a basis of the dual Hilbert space $V^*$.

This is related to but different from the sort of [[dual basis]] applicable generally to [[projective modules]].

\hypertarget{double_duals}{}\subsection*{{Double duals}}\label{double_duals}

The \textbf{[[double dual]]} of $V$ is simply the dual of the dual of $V$. There is a [[natural transformation]] from $V$ to its double dual:

\begin{displaymath}
\hat{x}(\lambda) = \lambda(x) ,
\end{displaymath}
for $x$ in $V$, $\lambda$ in $V^*$, and consequently $\hat{x}$ in $V^{**}$.

The space $V$ is called \textbf{[[reflexive Banach space|reflexive]]} if this natural transformation is an [[isomorphism]]. The reflexive spaces include all finite-dimensional vector spaces or modules over [[fields]] or [[division rings]], as well as all [[Hilbert spaces]], the [[Lebesgue spaces]] $L^p$ over a [[localizable measure|localisable measure space]] for $1 \lt p \lt \infty$, and others.

\hypertarget{dual_spaces_as_dual_objects}{}\subsection*{{Dual spaces as dual objects}}\label{dual_spaces_as_dual_objects}

A dual vector space is a [[dual object]] in the [[monoidal category]] [[Vect]] equipped with its [[tensor product]] monoidal structure.

In general, the duality between $V$ and $V^*$ does \emph{not} make $Vect$ into a [[monoidal category with duals]]. However, if we restrict to spaces of [[F-finite|finite]] [[dimension]], then we get a [[compact category]]; finite-dimensional [[Hilbert spaces]] form a $\dagger$-[[dagger-compact category|compact category]], which is very nice indeed.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{DualityBetweenVectorsAndCovectors}\hypertarget{DualityBetweenVectorsAndCovectors}{}
\textbf{(duality between [[vectors]] and [[covectors]])}

\begin{itemize}%
\item [[tensor|Tensors]] in finite-dimensional [[differential geometry]] make heavy use of the duality between [[tangent spaces]] and [[cotangent spaces]]. That [[infinite-dimensional manifold|infinite-dimensional differential geometry]] is much harder is largely because this duality is no longer perfect.

\end{itemize}
\end{example}
\begin{example}
\label{VonNeumannAlgebras}\hypertarget{VonNeumannAlgebras}{}
\textbf{([[von Neumann algebras]])}

A [[von Neumann algebra]] (abstractly) is precisely a $C^*$-[[C-star-algebra|algebra]] whose underlying [[Banach space]] is the dual space of some (other) Banach space. One may equivalently \emph{define} a von Neumann algebra as a Banach space together with a $C^*$-algebra structure on its dual space (except that the [[morphisms]] go the other way, so one is more directly defining a noncommutative [[measurable space]], along the lines of [[noncommutative geometry]]).

\end{example}
\begin{example}
\label{SpacesOFDistributionsAsDualVectorSpaces}\hypertarget{SpacesOFDistributionsAsDualVectorSpaces}{}
\textbf{(spaces of [[distributions]])}

The standard [[topological space|topology]] on the spaces $\mathcal{D}'$ of [[distributions]] is the dual space topology according to def. \ref{LinearDualOfATopologicalVectorSpace}.

(e.g. \hyperlink{Hoermander90}{H\"o{}rmander 90, p. 38})

\end{example}
See \emph{[[Riesz representation theorem]]} for more examples from [[functional analysis]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[dual space]]


\item [[dual vector bundle]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Discussion in the context of [[distributions]] includes

\begin{itemize}%
\item [[Lars Hörmander]], section 2.1 of \emph{The analysis of linear partial differential operators}, vol. I, Springer 1983, 1990

\end{itemize}
[[!redirects dual vector space]] [[!redirects dual vector spaces]]

[[!redirects dual module]] [[!redirects dual modules]]

[[!redirects left dual module]] [[!redirects left dual modules]] [[!redirects right dual module]] [[!redirects right dual modules]]

[[!redirects dual topological vector space]] [[!redirects dual topological vector spaces]] [[!redirects dual Banach space]] [[!redirects dual Banach spaces]] [[!redirects dual Hilbert space]] [[!redirects dual Hilbert spaces]]

[[!redirects alebraic dual space]] [[!redirects alebraic dual spaces]] [[!redirects alebraic dual]] [[!redirects alebraic duals]]

[[!redirects topological dual space]] [[!redirects topological dual spaces]] [[!redirects topological dual]] [[!redirects topological duals]]

[[!redirects linear dual]] [[!redirects linear duals]]



\end{document}