ABSTRACT
For the study of duality theory, there are mainly three fundamental principles: The first one is to consider the duality between locally convex topology and bornology: namely to a given dual pair <X, Y>, any locally convex topology (resp. convex bornology) on either X or Y, consistent with <X, Y>, corresponds by polarity a convex bornology (resp. locally convex topology) on the other space; this idea is extremely convenient since, in practice, it is much easier to construct convex bornologies than locally convex topologies satisfying some given properties. The second one is to consider the duality between equicontinuity and boundedness for an G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2736.tif"/> –topology; namely for a given LCS (X, P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2737.tif"/> ), one compares two natural convex bornologies on X'; the equicontinuous bornology M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2738.tif"/> equ( P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2739.tif"/> ) and the von Neumann bornology M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2740.tif"/> von( G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2741.tif"/> °), this leads to the Banac–Steinhaus theorem and the notions of barrelledness and the infrabarrelledness. The final one is to consider the duality between locally bounded maps and the continuity of their dual maps; this is one of the fundamental operations in Analysis. This section is devoted to a study of these three fundamental principles.
