Talk:Uniform convergence

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Thanks!Was a great help.... Stephan

Theorem assumption too strong?

In the section of theorems relating to the interchanging of limits, specifically the one pretaining to derivatives, I don't believe that uniform convergence of the original sequence of functions is necessary. It should be that if the sequence of functions {f_n(x)} converges pointwise and the sequence of derivatives {f_n'(x)} of the original sequence converges uniformly, then the original limit function f(x) = lim f_n(x) is differentiable and its derivative is equal to the limit of the sequence of the derivatives of the original sequence, f'(x) = lim f_n'(x). --anon

You are right. But I would prefer to leave it the way it is, it is more straightforward that way.

By the way, if the interval on which f_n are defined is bounded, it is enough for the sequence {f_n(x)} to be convergent just at one point. This together with the assumption that {f_n'(x)} converges uniformly, will imply that f_n converges uniformly to a function f and that f'(x) = lim f_n'(x). Oleg Alexandrov 02:37, 14 September 2005 (UTC)

Dirichlet or Seidel?

The article says:

Dirichlet then analyzed Cauchy's proof and found the mistake: the notion of pointwise convergence had to be replaced by uniform convergence.

On the other hand, infinite series says:

The theory of uniform convergence was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it successfully were Stokes and Seidel (1847-48).

Imre Lakatos (Proofs and Refutations, p. 135) also credits Stokes and Seidel, citing the following papers

• P. L. Seidel (1847). "Note über eine Eigenschaft der Reihen, welche Discontinuirliche Functionen Darstellen". Abhandlungen der Mathmatisch-Physikalischen Klasse der Königlich Bayerischen Akademie der Wissenschaften. 5: 381–393.
• G. Stokes (1848). "On the Critical Values of the Sums of Peroidic Series". Transactions of the Cambridge Philosophical Society. 8: 533–583.

Should this article be corrected? Gdr 22:26, 7 August 2006 (UTC)

Uniform convergence of the exponential function

I added in the example of uniform convergence of the exponential function, but now I'm not so sure that my argument was valid. Every time we change the radius of the disc D_R, we change the speed of convergence, because we're bounding all of the points by a different radius. I don't feel comfortable stating that the series is uniformly convergent for all z in the complex plane, at least based on the definition of uniform convergence in this article. What does anyone else think of this? —Preceding unsigned comment added by Bdforbes (talk • contribs) 02:14, 8 November 2008 (UTC)

For real numbers, the exponential function is uniformly convergent on any bounded interval [A, B], with A < B, which is the usual case. Now we can make the interval [A, B] as large as we please, as long as it is finite in length. So, don't let A be minus infinity, or B be plus infinity, and everything is fine.

So let's just consider B. This situation is a somewhat subtle. You can make B as large as you choose, but you can't let B be infinity.
What's the difference? For practical purposes, there isn't any difference. When mathematician say that the interval of convergence is infinite, that is just a shorthand way of saying, "You can make the interval of convergence as large as you will ever need."

For complex numbers, the exponential function is uniformly convergent on any bounded disk of radius R, with 0 < R. Now we can make the radius R as large as we please, as long as it is finite in length. So, don't let R be infinity, and everything is fine.

This situation is a somewhat subtle. You can make R as large as you choose, but you can't let R be infinity. What's the difference? For practical purposes, there isn't any difference. When mathematician say that the radius of convergence is infinite, that is just a shorthand way of saying, "You can make the radius of convergence as large as you will ever need."

The same thing applies for the functions sine(z) and cosine(z), but there comes a problem with tan(z) because this function has singularities in it, and it is not a continuous function.

Some other functions have extremely large radii of convergence, such a many of the Bessel functions. You really need to check out if the functions have singularities in them. For example, Log(z) has a very nasty singularity in it - a branch point - and it has a small radius of convergence.
98.67.106.59 (talk) 20:04, 4 August 2012 (UTC)

Image request

To someone with some graphics expertise: this article would benefit from an image of a uniform ε-neighborhood of a given function. (Just a "tube" of diameter 2ε around the graph of a function.) I would do it myself, but I'm pretty hopeless with graphics. Sławomir Biały (talk) 12:32, 18 October 2010 (UTC)

History Section

I wanted to add a bit to the history section to give a few more details. Also I want to rework it to remove weasel words such as "Some historians", but I thought I would state my intention on the talk page first. Thenub314 (talk) 00:44, 17 December 2011 (UTC)

Incorrect Statement in the Introductory Paragraph

The second sentence in this Wikipedia article says "A sequence {fn} of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x." I was under the impression that the speed of convergence may depend on x but is bounded, i.e. f doesn't converge arbitrarily quicker at certain points than at others. Can someone confirm this? And if so that statement should probably be changed. — Preceding unsigned comment added by Zaubertrank (talk • contribs) 04:16, 3 March 2012 (UTC)

"Speed of convergence" is a questionable concept. Mathematical things don't really have "speeds", even though they can be used to describe speeds in physics.
98.67.106.59 (talk) 20:11, 4 August 2012 (UTC)

No, speed of convergence is a valid concept (though it is better to say "rate", not "speed"). But indeed, the phrase "speed...does not depend on x" is somewhat misleading. On the other hand, "rate...is bounded" is hardly clear enough. Boris Tsirelson (talk) 20:19, 24 April 2014 (UTC)

Maybe this equivalent formulation could help: $|f_{n}(x_{n})-f(x_{n})|\to 0$ for every sequence $(x_{n})_{n}.$ Boris Tsirelson (talk) 20:30, 24 April 2014 (UTC)

Uniform convergence of analytic functions in the complex plane

My textbook on complex analysis states that if a sequence of analytic functions converges uniformly on a region of the complex plane, then the limit of the sequence is analytic in that region. This seems an important result (it shows that complex-differentiable functions are better behaved than real-differentiable functions) yet this article doesn't mention it at all. Perhaps it should be added under a new heading called "To analycity"? — Preceding unsigned comment added by JacekW (talk • contribs) 09:05, 27 January 2013 (UTC)

Almost uniform convergence

Ridiculously, this section says "converges uniformly almost everywhere", and then the opposite: "does not mean that the sequence converges uniformly almost everywhere as might be inferred from the name". Boris Tsirelson (talk) 14:06, 24 April 2014 (UTC)

To be honest I am not familiar with the this use of the phrase "almost uniform convergence". I glanced in a couple of texts hoping to find a reference and straighten this section out, the only one I could get my hands on that seems to discuss what this section wants to mention was "Real Mathematical Analysis" by Pugh. In an exercise on Egorov's theorem, he defines "nearly uniform convergence" to be what this article wants to define as "almost uniform convergence" and he uses the phrase almost uniform convergence to mean uniform convergence almost everywhere. But I am personally starting to think this is not important enough an idea to merit mentioning. I suggest we nuke the section. Thenub314 (talk) 15:45, 24 April 2014 (UTC)

"a property holds almost everywhere if the set of elements for which the property does not hold is a set of measure zero" (a quote from the article "Almost everywhere" linked from here). It is implicitly assumed that "the property" if defined (either holds or does not hold) for each point (of a measure space). Therefore the phrase "converges uniformly almost everywhere" is neither true nor false; rather it is meaningless, just like the phrase "converges uniformly at this point". Uniformity is about sets, not about points.

But Egorov's theorem deserves to be mentioned. "Nearly uniform convergence" sounds good. The link to "Almost everywhere" is erroneous. Boris Tsirelson (talk) 20:06, 24 April 2014 (UTC)

Agreed, I will try my hand at re-writing it. Thenub314 (talk) 20:11, 24 April 2014 (UTC)

I had the same problem as Boris (the section seemed to be saying contradictory things) so I grabbed my copy of Bartle ("The Elements of Integration and Lebesgue Measure") and edited the section. I confirmed the definition given there indeed has the implications mentioned (Egorov's theorem, AE convergence, convergence in measure) Shaun H. 71.2.44.232 (talk) 03:02, 3 December 2014 (UTC)

Nice; thank you.

Now I only bother about "Egorov's theorem guarantees that on a finite measure space..."; it is meant a space with a finite measure, of course, but could the reader think that a finite space (with measure) is meant? That is, only finitely many points? Boris Tsirelson (talk) 07:14, 3 December 2014 (UTC)

Recent edits by SergeyLiflandsky

1. Applications -> To continuity: is f continuous on S or I?

2. Proof: "Since f_n is continuous at x₀..." - which n is meant? If it is an arbitrary n, then $\delta (\varepsilon )$ is rather $\delta (n,\varepsilon ).$

Boris Tsirelson (talk) 06:03, 26 July 2015 (UTC)

Centralized discussion on proofs

See WT:MATH#Proofs, revisited — Arthur Rubin (talk) 17:58, 29 September 2015 (UTC)

However, the discussion of whether the proof is correct and placed in (a) correct section is not really discussed there. — Arthur Rubin (talk) 18:03, 29 September 2015 (UTC)

"Cauchy's wrong theorem"

There is some controversy on whether Cauchy actually made a mistake here or not. See the following for some lively discussion: one, two, three, four. Double sharp (talk) 08:56, 22 May 2020 (UTC)

In the section "Defintion", there may need more explanation of 𝑓 𝑛 ⇉ 𝑓 ⟺ 𝑑 ( 𝑓 𝑛 , 𝑓 ) → 0 and 𝐵 ( 𝑥 , 𝑟 ) ∩ 𝐸.

Hi.

In the section "Definition" of this article "Uniform convergence", there is the below (quoted here) description at the end of the section, and I think ${\textstyle d(f_{n},f)\to 0}$ in ${\textstyle f_{n}\rightrightarrows f\iff d(f_{n},f)\to 0}$ needs to have an explanation of "n goes to infinity".

And ${\textstyle B(x,r)}$ in ${\textstyle B(x,r)\cap E}$ seems a ball in a metric space but such identity is not mentioned here. Thus, it also needs to be explained.

I would like to get opinion of the abovementioned improvements. Are they OK to be implemented in this section?

"Symbolically,

$f_{n}\rightrightarrows f\iff d(f_{n},f)\to 0$.

The sequence $(f_{n})_{n\in \mathbb {N} }$ is said to be locally uniformly convergent with limit $f$ if $E$ is a metric space and for every $x\in E$, there exists an $r>0$ such that $(f_{n})$ converges uniformly on $B(x,r)\cap E.$ It is clear that uniform convergence implies local uniform convergence, which implies pointwise convergence." Goodphy (talk) 12:45, 1 March 2026 (UTC)

I would like to get opinion of the abovementioned improvements. Are they OK to be implemented in this section? Please say precisely what you would like to change it to. (In principle changes along those lines would be okay, but it is not possible from your description to know if the actual edit you want to make is okay.) --JBL (talk) 00:26, 2 March 2026 (UTC)

Hi JBL.

Replacing "d(f_n,f) → 0" with "d(f_n,f) → 0 as n → ∞" and mentioning "where B(x,r) is a ball centered at x with the radius r" are something improving this article's section "Definition" in my prospective. Goodphy (talk) 01:56, 2 March 2026 (UTC)

Hi Goodphy, both of these are fine, although please match the local formatting of mathematics, and the second one would be more idiomatic if you write "where B(x, r) is the ball centered at x with radius r". (The first one is perhaps not necessary -- every single limit in this article is a limit of a sequence, hence a limit as $n\to \infty$ -- but there's no strong reason not to do it.) --JBL (talk) 23:26, 2 March 2026 (UTC)

Hi.

I accepted your point that mentioning "n goes to ∞" (my 1st suggestion) may be necessarily (as it has been assumed through this article), so I tried to implement my 2nd suggestion of mentioning B as a ball in the definition of "locally uniformly convergence". However, I found that there are two balls; an open ball and a closed ball.

I found other resources saying neighborhood instead of a ball in the definition. Because I'm not sure which ball is used in this definition, I decided not to implement the suggestion to this article at this moment.

I hope other people would clarify this in the near future. Goodphy (talk) 11:32, 8 March 2026 (UTC)

The statement is true in either meaning (because every open ball with positive radius contains a closed ball with positive radius and vice-versa, or in other words because both kinds of balls in a metric space are topological neighborhoods). --JBL (talk) 17:34, 8 March 2026 (UTC)

HI JBL.

I just added the 2nd suggestion to the article by considering your advice above. I still think that a closed ball with the radius r contains the open ball with the same radius, but the opposite is not true, so "open" or "closed" needs to be specified, but such confusion may not be a strong driver to block mentioning what B(x,r) is. Goodphy (talk) 09:06, 12 March 2026 (UTC)

$C^{0}$ and $C^{1}$ topologies

In mathematical physics, it is common to speak of the $C^{0}$ and $C^{1}$ topologies, which are understood to mean the compact-open topology generated by uniform convergence of continuous functions (in the first case) and uniform convergence of the function and its first derivative in the second case (viz the Sobolev space $W^{1,\infty }$). I'm writing here because I want to wikilink some physics articles to something that defines these these two, but cannot find an appropriate linkage. Neither this article nor compact-open nor Sobolev mention or define $C^{0}$ and $C^{1}$. The compact-open article is great for math students, but is a bottomless black hole for physics students, who simply will not have the desire to plumb the depths of the implications of what is written there (even though clearly we can map delta-epsilon type proofs into there).

So I am asking for paragraphs or an article that (1) define $C^{0}$ and $C^{1}$ as spaces of uniform convergence of continuous resp. differentiable functions (2) use the word "topology", pointing at compact-open as the "general definition" (3) point out that these are the appropriate corresponding Banach and Sobolev spaces (4) are accessible to undergrads with a light background in mathematics. (For example, physics students ... and their professors ... will have a general if sometimes vague idea of what a topology is, and will have bumped into Hausdorff and Alexandrov interval topology, but will be reticent to dig deeper into formal complexities and definitions that are commonplace in mathematics. Why? Because they are already struggling with something else complicated. For example, globally hyperbolic spacetime throws around the symbols $C^{0}$ and $C^{1}$ like you're supposed to "just know what they mean". I just now moments ago wikilinked $C^{0}$ to this article. Unfortunately, this article never defines the symbol $C^{0}$ so that's losing. I could link $C^{1}$ to Sobolev, but that article also does not define this notation. I don't quite know where else to begin.

(p.s. here's another example from mathematical physics: $C^{0}$ symplectic topology which demonstrates some surprising rigidity properties of $C^{0}$ for Hamiltonian systems. It's actually kind of cool. Not something one might guess from saying "oh its just uniform convergence".) ~2026-15315-06 (talk) 22:35, 11 March 2026 (UTC)