What do I need to know to understand Uniform convergence?

[MAGNIBORO]

Hi, I started to study the function of Weierstrass (https://en.wikipedia.org/wiki/Weierstrass_function)
And in one part says that the sum of continuous functions is a continuous function.
i understand this but the Limiting case is a different history depend of the convergence, so what i need to know to self-study properly the uniform convergence, pointwise convergence, etc?

 

[chiro]

Hey MAGNIBORO.

Get an introductory book on topology that is self-contained [they tell you this sort of thing in the preface and/or initial chapters].

Real analysis requires a proper undergraduate sequence to have the intuition before you go into all the formalities.

 

[MAGNIBORO]

Hey MAGNIBORO.

Get an introductory book on topology that is self-contained [they tell you this sort of thing in the preface and/or initial chapters].

Real analysis requires a proper undergraduate sequence to have the intuition before you go into all the formalities.

ok thanks , I will have to continue studying the theme

 

[Stephen Tashi]

so what i need to know to self-study properly the uniform convergence, pointwise convergence, etc?

You need to understand the formal "epsilon-definition" of limits and the use of the logical quantifiers "there exists" and "for each". (e.g. https://www.physicsforums.com/threads/punctual-and-uniform-convergence.894339/#post-5629603 )

 

[FactChecker]

You need to study a reference to be precise, but here is a rough description:

Absolute[CORRECTION: Uniform]: For any difference you specify, no matter how small (say ε=0.001), there is an integer N (say 1000), where for all i≥N, |f_i(x) - f(x)| < ε for all x
Pointwise: Given an x, for any difference you specify, no matter how small (say ε=0.001), there is an integer N (say 1000), where for all i≥N, |f_i(x) - f(x)| < ε for that particular x

You can see that pointwise convergence is much weaker than absolute[correction: uniform]. In fact, if the series is only pointwise convergent, where may be no time where the series[correction: sequence] gets close to the limit at every value of x. The series f_i(x) = x^1/n is only pointwise convergent to f(x) ≡ 1.0 on R-{0}. You can always find x large enough to get keep f_i far from 1.0.

 

[S.G. Janssens]

You need to study a reference to be precise, but here is a rough description:

Absolute: For any difference you specify, no matter how small (say ε=0.001), there is an integer N (say 1000), where for all i≥N, |f_i(x) - f(x)| < ε for all x
Pointwise: Given an x, for any difference you specify, no matter how small (say ε=0.001), there is an integer N (say 1000), where for all i≥N, |f_i(x) - f(x)| < ε for that particular x

You can see that pointwise convergence is much weaker than absolute. In fact, if the series is only pointwise convergent, where may be no time where the series gets close to the limit at every value of x. The series f_i(x) = x^1/n is only pointwise convergent to f(x) ≡ 1.0 on R. You can always find x large enough to get keep f_i far from 1.0.

While the essence of what you say is correct, I have three remarks.

1. Absolute convergence is not the same as uniform convergence. (The former talks about series. The latter talks about sequences or series, where in the second case the series is interpreted as a sequence of partial sums.)

2. The $f_i$ in your example constitute a sequence on $\mathbb{R}$, not a series.

3. This sequence does not converge to $1$ at $x = 0$.

 

[FactChecker]

While the essence of what you say is correct, I have three remarks.

1. Absolute convergence is not the same as uniform convergence. (The former talks about series. The latter talks about sequences or series, where in the second case the series is interpreted as a sequence of partial sums.)

2. The $f_i$ in your example constitute a sequence on $\mathbb{R}$, not a series.

3. This sequence does not converge to $1$ at $x = 0$.

I stand corrected. Thanks. I got sloppy. I should not try to answer these while watching a football game.

 

[MAGNIBORO]

Thank you all for your comments, For these I understood the epsilon definition like make the difference or error between |fi(x) - f(x)| Arbitrarily small
I need to study more the epsilon definition and the sup and inf limits.
Many times i saw them And I guess As it seems is the key to many things in math

 

[S.G. Janssens]

Thank you all for your comments, For these I understood the epsilon definition like make the difference or error between |fi(x) - f(x)| Arbitrarily small
I need to study more the epsilon definition and the sup and inf limits.
Many times i saw them And I guess As it seems is the key to many things in math

My recommendation would be that you get a good introductory analysis book, such as Introduction to Real Analysis (4th edition) by Bartle and Sherbert. Pointwise and uniform convergence of sequences of functions is discussed in Section 8.1 there. (Indeed, this requires you to work through more foundational material first, but that need not be a bad thing.)

I would not start by reading a topology book, although some topology books (particularly those discussing function spaces) will address uniform convergence as well. (Bartle and Sherbert give a glimpse of topology in the final Chapter 11 of their book.) In my opinion, it is more natural to first understand the analytical definitions and then see how some of their implications can in turn be taken as defining properties in a topological context.