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 by a bilinear form. Using the bilinear form, 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.
We say puts and in duality.
We call two elements and orthogonal if
We call two sets and orthogonal if any two elements of and are orthogonal.
A vector space together with its algebraic dual and the bilinear form defined as
forms a dual pair.
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
form a dual pair.
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.