What is the difference between pointwise and normal convergence?

Question

I am taking a basic complex analysis course and I'm trying to understand the differences between different forms of convergence.

Specifically, I am trying to distinguish normal convergence from pointwise convergence. I searched around for a similar question, but I was only able to find a comparison between normal convergence and uniform convergence.

Do normal convergence and pointwise convergence imply the same conditions? If not, what is the difference? I'm basically just trying to gain some intuition here to better understand the course material.

Thank you in advance!

Answer 1

Take for example $f_n: [0,1]\rightarrow [0,1],\quad x\mapsto x^n$. Then $f_n$ converges pointwise to $\mathbb{1}_{1}$ (with $\mathbb{1}_{1}(1)=1$ and $\mathbb{1}_{1}(x)=0\ \forall x\neq 1$). But the $f_n$ don't converge uniform to this function, because $f_n\left(\frac{1}{\sqrt[n]{2}}\right)=\frac{1}{2}\ \forall n\in \mathbb{N}$ whereas $\mathbb{1}_{1}\left(\frac{1}{\sqrt[n]{2}}\right)=0$.

Answer 2

"Normal" convergence is the same as pointwise convergence. Uniform convergence is generally a stronger condition, in the sense that uniform convergence implies normal (pointwise) convergence. On a compact domain, the converse holds as well, and normal (pointwise) convergence implies uniform convergence.