Let be a collection of continuous linear maps from a Banach space into a normed space . Then the following are equivalent:
(1)
(2) there exists such that with and
(3) there exists such that and
Let be a collection of continuous linear maps from a Banach space into a normed space . Then the following are equivalent:
(1)
(2) there exists such that with and
(3) there exists such that and
That is the statement of the Banach–Stainhaus theorem, and you can find a proof of it here. Is that what you wanted?