Let will be a pairing such that distinguishes the points of and let be a locally convex topology on . Then is compatible with the pairing if and only if is a polar topology determined by some collection of -compact disks that cover
Suppose that is a Hausdorff locally convex space with continuous dual space and consider the canonical duality If is any topology on that is compatible with the duality on then the bounded subsets of are the same as the bounded subsets of
Let be a pairing over the field Then the continuous dual space of is Furthermore, If is a continuous linear functional on then there exists some such that ; if such a exists then it is unique if and only if distinguishes points of * Note that whether or not distinguishes points of is not dependent on the particular choice of The continuous dual space of may be identified with the quotient space where * This is true regardless of whether or not distinguishes points of or distinguishes points of
Mackey–Arens theorem II
Let will be a pairing such that distinguishes the points of and let be a locally convex topology on Then is compatible with the pairing if and only if
Proposition
Let will be a pairing such that distinguishes the points of and let be a topology of the pair. Then a subset of is a barrel in if and only if it is equal to the polar of some -bounded subset of
Mackey's theorem
Dual system
Assume that distinguishes points of and is a linear map. Then the following are equivalent: # is weakly continuous ; # ; # the transpose of is well-defined. If is weakly continuous then * is weakly continuous, meaning that is continuous; * the transpose of is well-defined if and only if distinguishes points of in which case
Let be a TVS with algebraic dual and let be a basis of neighborhoods of at the origin. Under the canonical duality the continuous dual space of is the union of all as ranges over .
Dual pair
Mackey–Arens theorem I
Theorem
Weak representation theorem