I Pointwise and Uniform Convergence

1. May 29, 2017

Silviu

Hello! Can someone explain to me in an intuitive way the difference between pointwise and uniform convergence of a series of complex functions $f_n(z)$? Form what I understand, the difference is that when choosing an "N" such that for all $n \ge N$ something is less than $\epsilon$, in the case of uniform convergence N can't depend on z but in the case of pointwise convergence it can. I saw an example for the function $f_n(z)=z^n$ which has uniform convergence if the domain is $D_{[0,a]}$ with $a<1$, but has pointwise convergence for $a=1$. I understand that the proofs involves different approaches and this is why they have different types of convergence, but in the end they converge to the same function $f(z)=0$. So how is the uniform convergence stronger than the other one (I know that pointwise is implied by uniform but not the other way around, but this still doesn't really give me a clear understanding). Thank you!

Last edited: May 29, 2017
2. May 29, 2017

hilbert2

If you have a function $f_n (x) = x^n$ on the half-open interval $[0,1[$, the sequence $f_1 (a),f_2 (a), f_3 (a), \dots$ will approach zero no matter what the number $a\in [0,1[$ is, but for any given $n$, no matter how large, there is some number $x\in [0,1[$ for which $f_n (x) = 0.5$ or any number in the interval $[0,1[$. So you can't limit the value of $f_n (x)$ to some interval $[0,c]$ where $c< 1$ by any finite choice of $n$.

3. May 29, 2017

lavinia

The main thing about uniform convergence that I know is that the uniform limit of continuous functions is continuous. The point wise limit may not be continuous.

4. May 29, 2017

Stephen Tashi

Are you talking about uniform convergence "of a complex function"? - or uniform convergence of a sequence of functions? It would be best to ask a precise question, even if you want an intuitive explanation.

5. May 29, 2017

Silviu

Sorry, I meant series of functions (anyway for a single function I guess it doesn't make a difference as there is no "n" to take the limit of, just z)

6. May 29, 2017

Stephen Tashi

I think you mean "sequence" of functions.

I don't what "it" refers to.

Do you understand the concept of uniform convergence for a sequence of real valued functions of real variable?