▁want ▁to ▁consider ▁Pon try agin ▁du ality ▁beyond ▁the ▁locally ▁compact ▁case . ▁El ena ▁Martín - Pe in ador ▁proved ▁in ▁ 1 9 9 5 ▁that ▁if ▁ ▁is ▁a ▁Haus dor ff ▁ab elian ▁topological ▁group ▁that ▁satisfies ▁Pon try agin ▁du ality , ▁and ▁the ▁natural ▁evaluation ▁pair ing ▁ ▁is ▁( joint ly ) ▁continuous , ▁then ▁ ▁is ▁locally ▁compact . ▁As ▁a ▁cor ollary , ▁all ▁non - loc ally ▁compact ▁examples ▁of ▁Pon try agin ▁du ality ▁are ▁groups ▁where ▁the ▁pair ing ▁ ▁is ▁not ▁( joint ly ) ▁continuous . ▁ ▁Another ▁way ▁to ▁general ize ▁Pon try agin ▁du ality ▁to ▁wider ▁classes ▁of ▁commut ative ▁topological ▁groups ▁is ▁to ▁end ow ▁the ▁dual ▁group ▁ ▁with ▁a ▁bit ▁different ▁topology , ▁namely ▁the ▁topology ▁of ▁uniform ▁convergence ▁on ▁totally ▁bounded ▁sets . ▁The ▁groups ▁satisfying ▁the ▁identity ▁ ▁under ▁this ▁assumption ▁are ▁called ▁s tere ot ype ▁groups . ▁This ▁class ▁is ▁also ▁very ▁wide ▁( and ▁it ▁contains ▁locally ▁compact ▁ab elian ▁groups ), ▁but ▁it ▁is ▁nar ro wer ▁than ▁the ▁class ▁of ▁reflect ive ▁groups . ▁ ▁Pon try agin ▁du ality ▁for ▁topological ▁vector ▁spaces ▁ ▁In ▁ 1 9 5 2 ▁Marian ne ▁F . ▁Smith ▁noticed ▁that ▁Ban ach ▁spaces ▁and ▁ref lex ive ▁spaces , ▁being ▁considered ▁as ▁topological ▁groups ▁( with ▁the ▁add itive ▁group ▁operation ), ▁satisfy ▁Pon try agin ▁du ality . ▁Later ▁B . ▁S . ▁Br ud ov