Talk:Uniform boundedness principle

Early comments[edit]

What the hell?! --JensMueller

Perhaps, the French professor has to think first then post, especially in a language as foreign to him as English.

What do you mean exactly? It was simply not true that

\|T\|\leq 2n/\|z\|.

--Bdmy (talk) 12:45, 28 April 2009 (UTC)[reply]

The orginal non-Latex text of the proof[edit]

For n = 1,2,3, ... let X_n = { x : ||T(x)|| ≤ n (∀ T ∈ F) } . By hypothesis, the union of all the X_n is X.

Since X is a Baire space, one of the X_n has an interior point, giving some δ > 0 such that ||x|| < δ ⇒ x ∈ X_n.

Hence for all T ∈ F, ||T|| < n/δ, so that n/δ is a uniform bound for the set F.

I am reverting to the non-latex version, because the latex version adds nothing. (It has actually subtracted something: a rather important "implies" sign.) See Wikipedia:How_to_write_a_Wikipedia_article_on_Mathematics. Andrew Kepert 22:46, 14 Mar 2005 (UTC)

consistency[edit]

The second theorem, about the pointwise limit, could be stated a bit better. The operators should be 'linear', no? I know this is probably implicity assumed, but this is not the place for implicit assumptions. And the article interchanges 'continuous' and 'bounded' when referring (presumably) to linear operators. While they might be the same in Banach Spaces (they are, no?), they are still separate ideas and are not always the same in all topological vector spaces. So let's pick one, and stick with it. Also, I like the non-latex version of the proof on the talk page better than the one on the main page. Lavaka 00:28, 21 September 2006 (UTC)[reply]

As the second theorem is a consequence of the first, I'd guess that the assumptions are the same: The T_n are linear operators from a Banach space to a normed vector space. -- Jitse Niesen (talk) 06:39, 6 October 2006 (UTC)[reply]

Looking back at the proof, the wording is very bizarre: "Besides for all z in X such ..." What does that mean? Lavaka 23:28, 13 March 2007 (UTC)[reply]

"Besides" means "furthermore" here. Anyway, I rewrote the proof in an attempt to clarify it; does that help? -- Jitse Niesen (talk) 04:20, 14 March 2007 (UTC)[reply]

choice?[edit]

Since the proof here uses choice, should there not be note about whether or not the theorem requires choice? (I was under the impression that it does not). 18.209.1.89 (talk) 14:24, 26 April 2010 (UTC)[reply]

Typo[edit]

I think there is a typo in the proof of the uniform boundedness principle, where all the epsilon should be replaced by epsilon^{-1}.

I agree. I changed it. Bdmy (talk) 16:50, 10 July 2013 (UTC)[reply]

On the case where $X$ is the zero space[edit]

In the statement of the theorem, it was claimed that $\sup \nolimits _{T\in F,\|x\|=1}\|T(x)\|_{Y}=\sup \nolimits _{T\in F}\|T\|_{B(X,Y)}$ . But if $X$ is the zero space, then $-\infty =\sup \nolimits _{T\in F,\|x\|=1}\|T(x)\|_{Y}\neq \sup \nolimits _{T\in F}\|T\|_{B(X,Y)}=0$ . EpicGuy4227 (talk) 10:16, 22 July 2020 (UTC)[reply]

I'd say that here, like in many other analogous cases, it is understood that the supremum is taken in the set of non-negative real numbers, where

\sup \emptyset =0

. pm a 12:22, 24 July 2020 (UTC)[reply]

Corollaries[edit]

  This article is very clear and well written and I would like to thank the authors. I did have a slight problem with the content at one place. In the first "Proof" in the "Corollaries" section the author starts by saying "Essentially the same as that of the proof that a pointwise convergent sequence of uniformly continuous functions on a compact set converges to a continuous function". But this is not true. Any continuous function on a compact set (in a Metric Space) is uniformly continuous, so this asserts that any pointwise convergent sequence of continuous functions converge to a continuous function.

  I think the author meant "equicontinuous" instead of "uniformly continuous" and the conclusion was meant to say "converges uniformly" rather than "converge to a continuous function" since the point of the proof is to establish uniform convergence and not simply the continuity of the limit. So I believe the statement should be:

  "Essentially the same as that of the proof that a pointwise convergent equicontinuous sequence of functions on a compact set converges uniformly".

  One other point, though a bit picky, is that the "Proof" is not a proof of the first "Corollary" but rather a proof of the statement following the corollary. I was briefly confused by this until I realized what was actually being proved. It might be better to add another corollary box for this statement, though I suppose this is mostly a matter of personal taste. Catgod119 (talk) 08:18, 30 December 2022 (UTC)[reply]

Yes, the right word would be "equicontinuous". In fact, such sentences as: "Proof: Essentially the same as (..)" should be avoided in these article, unless there is a precise reference to link to. But in fact the equicontinuity is not needed for the proof of the Corollary as stated; the proof is just: A point-wise converging sequence $(T_{k})_{k}$ of linear bounded operators on a Banach space is bounded, because the uniform boundedness theorem applies. By point-wise convergence, linearity and the bound $\|T_{k}x\|\leq M\|x\|\forall x\in X$ hold true for the limit map, which is therefore a bounded linear operator. As you observed, the consequence of the equicontinuity is needed instead to deduce another consequence: the uniform convergence on compact sets, which is however not mentioned in the statement of the Corollary. I wouldn't say this article is clear and well written. There is still much work to be done.pm a 09:27, 31 December 2022 (UTC)[reply]