See here: Alan D. Sokal, A really simple elementary proof of the uniform boundedness theorem, Amer. Math. Monthly 118, 450-452 (2011)
https://arxiv.org/abs/1005.1585, http://dx.doi.org/10.4169/amer.math.monthly.118.05.450
The proof I'm referring to can be found here.
I have worked through this proof and am literally at the last line of the UBP proof but am stuck. I have two questions:
Firstly, how does he obtain the inequality $|| x - x_n || \leq \frac{1}{2}3^{-n}$? I can get $|| x - x_n || \leq (6)3^{-n}$ via ...
The uniform boundedness principle of Functional Analysis is a very important application of the Baire Category Theorem.
Added: (t.b.) See also Sokal's A really simple elementary proof of the uniform boundedness theorem for a proof without Baire.
The uniform boundedness principle of Functional Analysis is a very important application of the Baire Category Theorem.
Added: (t.b.) See also Sokal's A really simple elementary proof of the uniform boundedness theorem for a proof without Baire.
