abstract duality: opposite category,
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.
The left dual module of is the right -module of left -module homomorphisms from to . The right dual module of is the left -module of right -module homomorphisms from to .
The dual space of is the topological vector space of continuous linear functionals on , equipped with the weak-* topology? (meaning the initial topology generated by the elements of , viewed as themselves linear functionals on ).
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 is an algebra over another rig , then any -module is also an -module, but the dual as a -module is different from the dual as an -module. So one may speak of the -dual or the dual over .
A topological vector space has an underlying discrete vector space, and these have different duals. So one speaks of the topological dual and the algebraic dual (respectively). If 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 extends to a contravariant functor.
for all in and in .
If is any basis of , then we can sometimes turn into a basis of the dual space .
We will begin with the definition of what might be the dual basis, cautioning that it is not always a basis:
Treating the basis as a family with index set , the dual basis is the family (with the same index set) such that
(the Kronecker delta).
Since is a basis of , this formula defines for each index , so exists; but in general there is no reason why should be a basis of . However, if has finite dimension, then is a basis of . If is a Hilbert space, and is a basis of in the Hilbert space sense (i.e., is a linearly independent set whose span is topologically dense in ), then also is a basis of the dual Hilbert space .
for in , in , and consequently in .
The space 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 over a localisable measure space for , and others.
In general, the duality between and does not make 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 -compact category, which is very nice indeed.
A von Neumann algebra (abstractly) is precisely a -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 -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.