. ▁ ▁Definition ▁ ▁A ▁topological ▁group ▁is ▁called ▁locally ▁compact ▁if ▁the ▁underlying ▁topological ▁space ▁is ▁locally ▁compact ▁and ▁Haus dor ff ; ▁the ▁topological ▁group ▁is ▁called ▁ab elian ▁if ▁the ▁underlying ▁group ▁is ▁ab elian . ▁Ex amples ▁of ▁locally ▁compact ▁ab elian ▁groups ▁include ▁finite ▁ab elian ▁groups , ▁the ▁integers ▁( both ▁with ▁the ▁discrete ▁topology , ▁but ▁also ▁with ▁the ▁usual ▁metric ), ▁the ▁real ▁numbers , ▁the ▁circle ▁group ▁T ▁( with ▁their ▁usual ▁metric ▁topology ), ▁and ▁also ▁the ▁p - adic ▁numbers ▁( with ▁their ▁usual ▁p - adic ▁topology ). ▁For ▁a ▁locally ▁compact ▁ab elian ▁group ▁G , ▁the ▁Pon try agin ▁dual ▁is ▁defined ▁to ▁be ▁the ▁group ▁ ▁of ▁continuous ▁group ▁hom omorph isms ▁from ▁G ▁to ▁the ▁circle ▁group ▁T . ▁That ▁is , ▁▁ ▁can ▁be ▁end owed ▁with ▁the ▁topology ▁given ▁by ▁uniform ▁convergence ▁on ▁compact ▁sets ▁( i . e ., ▁the ▁compact - open ▁topology , ▁view ing ▁ ▁as ▁a ▁subset ▁of ▁the ▁space ▁of ▁all ▁continuous ▁functions ▁from ▁ ▁to ▁). ▁ ▁For ▁example , ▁ ▁The ▁Pon try agin ▁du ality ▁theorem ▁ ▁Theorem . ▁There ▁is ▁a ▁canonical ▁isomorphism ▁ ▁  ▁between ▁any ▁locally ▁compact ▁ab elian ▁group ▁ ▁and ▁its ▁double ▁dual . ▁ ▁Can on ical ▁means ▁that ▁there ▁is ▁a ▁naturally ▁defined ▁map ▁  ; ▁more ▁important ly , ▁the ▁map ▁should ▁be ▁fun ctor ial ▁in ▁. ▁The ▁canonical ▁isomorphism ▁is ▁defined ▁on ▁ ▁as ▁follows :