Dual pair
In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear map to the base field.
A common method in functional analysis, when studying normed vector spaces, is to analyze the relationship of the space to its continuous dual, the vector space of all possible continuous linear forms on the original space. A dual pair generalizes this concept to arbitrary vector spaces, with the duality being expressed as a bilinear map. Using the bilinear map, semi norms can be constructed to define a polar topology on the vector spaces and turn them into locally convex spaces, generalizations of normed vector spaces.
Definition
A dual pair[1] is a 3-tuple consisting of two vector spaces
and
over the same field
and a bilinear map
with
and
We call the duality pairing, and say that it puts
and
in duality.
When the two spaces are a vector space (or a module over a ring in general) and its dual
, we call the canonical duality pairing
the natural pairing.
We call two elements and
orthogonal if
We call two sets and
orthogonal if each pair of elements from
and
are orthogonal.
Example
A vector space together with its algebraic dual
and the bilinear map defined as
forms a dual pair.
A locally convex topological vector space space together with its topological dual
and the bilinear map defined as
forms a dual pair. (To show this, the Hahn–Banach theorem is needed.)
For each dual pair we can define a new dual pair
with
A sequence space and its beta dual
with the bilinear map defined as
form a dual pair.
Comment
Associated with a dual pair is an injective linear map from
to
given by
There is an analogous injective map from to
.
In particular, if either of or
is finite-dimensional, these maps are isomorphisms.