The above pdf contains some notes on the uniform boundedness principle of functional analysis. The majority of notes is from the perspective of proving the existence of a continuous function whose Fourier series diverges at a single point. By seeing the concepts in the same light, my goal was to gain some intuition for both.
I’d like to thank two of my fellow graduate students, Chris Gartland and Hadrian Quan, for their useful suggestions.
My fellow graduate student, Martino Fassina, showed me this ridiculously simple proof of uniform boundedness.
. It is easy to see that no sub-sequence of 
can weak-star converge, yet Banach-Alaoglu asserts that the unit ball of 
is weak-star compact. So what is the problem?
and a closed subspace 
. Elementary considerations assert that the open unit ball of 
equipped with the 
with different topologies.