(also nonabelian homological algebra)
abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
A 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 duals; and there are other notions of dual space in physics.
Let $K$ be a field (or any commutative rig), and let $V$ be a vector space (or module) over $K$.
The dual space or 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).
More generally, let $K$ and $L$ be rings (or rigs) not assumed commutative, and let $V$ be a $K$-$L$-bimodule.
The left dual module of $V$ is the right $K$-module $^*V$ of left $K$-module homomorphisms from $V$ to $K$. The right dual module of $V$ is the left $L$-module $V^*$ of right $L$-module homomorphisms from $V$ to $L$.
Now let $V$ be a topological vector space over the ground field $K$.
The dual space of $V$ is the topological vector space $V^*$ of continuous linear functionals on $V$, equipped with the weak-* topology (meaning the initial topology generated by the elements of $V$, viewed as themselves linear functionals on $V^*$).
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:
If the rig $K$ is an 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 $K$-dual or the dual over $K$.
A topological vector space $V$ has an underlying discrete vector space, and these have different duals. So one speaks of the topological dual and the algebraic dual (respectively). If $V$ is considered with several different topologies (say called ‘weak’ and ‘strong’), then one may speak of the weak dual and the strong dual (etc).
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.
The operation $V \mapsto V^*$ extends to a contravariant functor.
The dual linear map or transpose map of a linear map $A\colon V\to W$, is the linear map $A^* = A^T\colon W^*\to V^*$, given by
for all $w$ in $W^*$ and $v$ in $V$.
This functor is, of course, the representable functor represented by $K$ as a vector space over itself (a line).
This construction is the notion of dual morphism applied in the monoidal category Vect with its tensor product monoidal structure.
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 might be the dual basis, cautioning that it is not always a basis:
Treating the basis $B$ as a family $(b_i)$ with index set $I$, the dual basis $B^*$ is the family $(b^i)$ (with the same index set) such that
(the Kronecker delta).
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 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.
The double dual? of $V$ is simply the dual of the dual of $V$. There is a natural transformation from $V$ to its double dual:
for $x$ in $V$, $\lambda$ in $V^*$, and consequently $\hat{x}$ in $V^{**}$.
The space $V$ is called 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 localisable measure space for $1 \lt p \lt \infty$, and others.
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 not make $Vect$ into a monoidal category with duals. However, if we restrict to spaces of finite dimension, then we get a compact category; finite-dimensional Hilbert spaces form a $\dagger$-compact category, which is very nice indeed.
A von Neumann algebra (abstractly) is precisely a $C^*$-algebra whose underlying Banach space is the dual space of some (other) Banach space. One may equivalently 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).
Tensors in finite-dimensional differential geometry make heavy use of the duality between tangent spaces and cotangent spaces. Infinite-dimensional differential geometry is much harder, largely because this duality is no longer perfect.
See Riesz representation theorem for examples from functional analysis.