# Original proof of Uniform Boundedness Principle (Banach Steinhaus) and related questions

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?

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I do not know Brezis' proof, but on complete spaces an immediately consequence of Baire Category Thm is the uniform boundedness (on an open ball) of continuous bounded real valued functions from your complete Space. With that claim it's easy to prove Banach-Steinhaus. Rudin's proof is beautiful and I think, sorry if I'm wrong, it's the most general one. – math Sep 22 '11 at 11:39
If you start with Baire Category Thm., where you need completeness (or locally compact Hausdorff) the claim above follows easily. I guess, then it's a natural question to ask if something like this is true on the whole space. – math Sep 22 '11 at 11:45
I agree that Rudin's proof must be close to the most general formulation, but I don't think it was the original proof. I also wonder wether the Baire Category theory made any appearance in the original proof... – Olivier Bégassat Sep 22 '11 at 11:58

Well, it is a good idea to look at the Wikipedia page first. There you'll find a link to the original article by Banach-Steinhaus which was motivated by the theory of Fourier series.

Concerning your question whether it was standard to apply the Baire category theorem at those times, very much so. The Polish school (e.g. Sierpinski, Lusin, Souslin, and of course Banach) applied it routinely, just look at their papers in the early volumes in Fundamenta Mathematicae.

Lastly, the proof in that paper you link to is quite a common gliding hump argument, which was recently published by Sokal in the Monthly. You'll find further historical remarks and discussion in his article.

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thank you for your answer. I did look it up on wikipedia first, but missed the link. I also found the paper you link to, and I like it very much! – Olivier Bégassat Sep 22 '11 at 13:12
@commenter: You'd created a duplicate account, and the fact that this new account did not have much rep meant that you were not able to include those links when you were logged in with that account. I've now merged it with your main account. If you have any trouble in the future, you can flag one of your own posts for moderator attention, and we'll help out. – Zev Chonoles Sep 27 '11 at 7:46
@Zev: Thanks a lot. I think it should be fine now. Yes, I had inadvertently created duplicate accounts before registering the current account. I checked using Google and it seems that all my answers are now collected under my current account. Thanks again! – commenter Sep 27 '11 at 7:54
@t.b. Thanks for editing in the links. – commenter Sep 27 '11 at 7:54

See here :arxiv.org/pdf/1005.1585

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In case the link goes stale, please give some details, or at least an overview, of what is gone over in the cited article. – robjohn Jan 8 '13 at 22:30