 A series can be called "locally normally convergent on "X"" if each point "x" in "X" has a neighborhood "U" such that the series of functions "ƒ""n" restricted to the domain "U"
is normally convergent, i.e.such that
where the norm formula_9 is the supremum over the domain "U".A series is said to be "normally convergent on compact subsets of "X"" or "compactly normally convergent on "X"" if for every compact subset "K" of "X", the series of functions "ƒ""n" restricted to "K"
is normally convergent on "K".