Problem of the Week 5

So I thought it would be fun to show some exercises here taken from the FA2 course I did last semester. Instead of writing it all out, I will just link the course page and label the exercise sheet. Exercise 5.4 from...

Solution to Problem of the Week 4

So, this is really late, the semester took its toll on me, and again nothing works. The first problem is an application of the Eberlain-Smulyan theorem; let be a countable dense subset of and note that for each there exists a subsequence such that is weakly...

Solution to Problem 3

So this problem comes from the functional analysis course I took last semester. Although simple, I thought was pretty cool. In the actual exercise they had broken it down into three parts, which I did not show in the problem post. Here I will show the steps as claims....

Principles: Uniform Boundedness

So last week we talked about the Baire Category Theorem, and this important tool will give us three equivalent theorems about the qualitative properties of operators on Banach spaces. Today we will begin with the first theorem, uniform boundedness. The proof of this...