A bounded linear operator is continuous so the inverse image of a closed set is closed.
I'm assuming that the statement of the Theorem is known to you but can write it out if needed.
The proof of the Uniform Boundedness Theorem(Banach-Steinhaus) uses the
following sets at the beginning of the proof:
A_n = {x in X: ||T_i(x)|| <= n for all i in I},where I is an indexing set, n is taken from natural numbers and where each T_i is a
bounded linear operator between a Banach Space X and a normed space Y.
How do we show that each A_n is closed?
Thanks