In mathematics, duality is the study of dual systems and is important in functional analysis.
Duality plays crucial roles in quantum mechanics because it has extensive applications to the theory of Hilbert spaces.
for the canonical evaluation map (defined below) so as to avoid confusion for readers not familiar with this subject.
is non-degenerate, which means that it satisfies the following two separation axioms: In this case
The definition of a subset being orthogonal to a vector is defined analogously.
To use bookkeeping that helps keep track of the anti-symmetry of the two sides of the duality, the absolute polar of a subset
[1] There is a consistent theme in duality theory that any definition for a pairing
") is defined as above, then this convention immediately produces the dual definition of "
Although it is technically incorrect and an abuse of notation, this article will adhere to the nearly ubiquitous convention of treating a pairing
which is called the evaluation map or the natural or canonical bilinear functional on
is a Hausdorff locally convex space) then this pairing forms a duality.
is a complex pre-Hilbert space with scalar multiplication denoted as usual by juxtaposition or by a dot
Importantly, the weak topology depends entirely on the function
is a pairing then the following are equivalent: The following theorem is of fundamental importance to duality theory because it completely characterizes the continuous dual space of
is endowed with the strong dual topology (and so is denoted by
(this topology is also called the strong bidual topology and it appears in the theory of reflexive spaces: the Hausdorff locally convex TVS
and it will be called reflexive if in addition the strong bidual topology
's transpose or adjoint is well-defined if the following conditions are satisfied: In this case, for any
By the conventions mentioned at the beginning of this article, this also defines the transpose of linear maps of the form
The following result shows that the existence of the transpose map is intimately tied to the weak topology.
There exist Banach spaces that are not weakly-complete (despite being complete in their norm topology).
is a Hausdorff locally convex TVS with continuous dual space
is a linear map between two Hausdorff locally convex topological vector spaces, then:[1] Let
[1] Some authors (e.g. [Trèves 2006] and [Schaefer 1999]) require that a topology of a pair also be Hausdorff,[2][8] which it would have to be if
(as was shown in the Weak representation theorem) and it is in fact the weakest such topology.
is equal to the intersection of all closed half spaces containing it.
[9] The above theorem implies that the closed and convex subsets of a locally convex space depend entirely on the continuous dual space.
Consequently, the closed and convex subsets are the same in any topology compatible with duality;that is, if
[1] The following theorem shows that barrels (i.e. closed absorbing disks) are exactly the polars of weakly bounded subsets.
In short, it states the bounded subsets are the same for any two Hausdorff locally convex topologies that are compatible with the same duality.