Can someone please provide me with any of the things listed below :
- a list of different proofs of (some version of) the Uniform Boundedness Principle (also known as the Banach Steinhaus theorem), I already know Rudin's proof that seems quite general, a proof in the case of Banach spaces found in Haïm Brezis' book on functional analysis (also based on Baire Category) and a different proof altogether (making no use of Baire Category) found here https://pantherfile.uwm.edu/kevinm/www/qtbook/notes/pub.pdf.
- if possible the orignial proof and formulation of the theorem.
The reason I ask is that I don't understand how people realised completeness was a necessary condition, also I wonder if using the Baire Category Theorem was standard in Banach's time. Finally, what motivating examples did Banach or Steinhaus consider?