Example
A vector space together with its algebraic dual and the bilinear form defined as
forms a dual pair.
A locally convex topological vector space space together with its topological dual and the bilinear form 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 form defined as
form a dual pair.
Read more about this topic: Dual Pair
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“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
—Marcel Proust (1871–1922)