1. ## Question about Uniform Boundedness Theorem

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

2. ## Re: Question about Uniform Boundedness Theorem

A bounded linear operator is continuous so the inverse image of a closed set is closed.

3. ## Re: Question about Uniform Boundedness Theorem

I'm probably missing something simple but A_n is not the inverse of T_i it is of ||T_i||?