# Difference between normal convergence, pointwise convergence and uniform convergence

Let be $$\left(f_k\right)_{k\in\mathbb{N}}$$ a sequence of functions, $$f_k:M\to Y$$, where $$(Y,\Vert\cdot\Vert_Y)$$ is a normed space.

Why do we sometimes talk about "normal" convergence and sometimes we refer to uniform convergence or pointwise convergence? Is it correct if I explain like this:

If we talk about $$\left(f_k\right)_{k\in\mathbb{N}}$$ as a sequence that lives in a function space then we simply refer to convergence and say $$\left(f_k\right)_{k\in\mathbb{N}}$$ converges to some element $$f$$ from this function space and write $$\lim\limits_{k\to\infty}f_k=f$$. However, if we look at a sequence of functions regardless its function space then we make the distinction between the two notions of convergence. So the distinction between convergence, uniform convergence and pointwise convergence comes from the different levels of abstractions.

EDIT:

Here an example to illustrate the idea:

Let's say the function space is equipped with the supremum norm, then in this case convergence of $$\left(f_k\right)_{k\in\mathbb{N}}$$ means that it is uniformly convergent (and vice versa). If the function space is equipped with some other norm then it could be the case that $$\left(f_k\right)_{k\in\mathbb{N}}$$ is uniformly convergent (and hence pointwisely convergent) but not convergent in the function space.

Is this explanation correct?

Maybe you have some other explanations which help to grasp the idea :)

• math.stackexchange.com/questions/597765/… Dec 20, 2020 at 13:07
• @GAUSS1860 I am not looking for an explanation of the difference between pointwise and uniform convergence. Dec 20, 2020 at 13:13
• Perhaps from the title it might look so. Dec 20, 2020 at 13:14

For example,$$f_n:(0,\infty)\to \mathbb{R}$$ given by $$f_n(x) = \frac{1}{nx}$$ is a sequence of functions that converges pointwise to 0. This is clear because if you pick any $$x_0\in (0,\infty)$$, then $$f_n(x_0) = \frac{1}{nx_0}$$ is just a sequence of real numbers clearly converging to $$0$$. But $$f_n$$ does not converge uniformly to $$0$$, because you can always find, for some $$\varepsilon$$, a $$\delta>0$$ such that for all $$x\in (0,\delta)$$, $$f_n(x) \geq \varepsilon$$. The idea is when $$x$$ is large then clearly $$f_n(x)=\frac{1}{nx}$$ will be small for all values of $$n\in\mathbb{N}$$. But when you shrink $$x$$ you have to increase $$n$$ for $$f_n(x)$$ to remain small and this relationship between $$x$$ and $$n$$ implies that the convergence is not uniform as it depends on what point we pick for us to determine the convergence, whereas if it were uniform then for large enough $$n$$, EVERY $$x$$ would satisfy $$f_n(x) <\varepsilon$$
• No I didn't mean pointwise convergence when I mentioned "normal" convergence. Let me explain it in other words: If you consider $\left(f_k\right)_{k\in\mathbb{N}}$ whose members are elements of a function space then this sequence can be convergent or not. In my opinion it makes no sense to talk about pointwise or uniform convergence at this level. However, as the elements of the sequence are of special nature in a sense that they are functions, we can apply different concepts to determine how convergence in the function space can be defined. Dec 20, 2020 at 12:44
• Yes exactly. I think I got it right now. If I consider the function space $(\mathcal{C}[a,b],\Vert\cdot\Vert_1)$, where $\Vert \cdot\Vert_{1}:= \int\limits_a^b|f(t)|dt$ then each sequence $\left(f_n\right)_{n\in\mathbb{N}}$ can be convergent with regard to $\Vert \cdot\Vert_{1}$. In this case pointwise and uniform convergence are irrelevant to decide if $\left(f_n\right)_{n\in\mathbb{N}}$ converges. However, I can still check independently from the question of convergence in $(\mathcal{C}[a,b],\Vert\cdot\Vert_1)$ if $\left(f_n\right)_{n\in\mathbb{N}}$ is pointwisely or uniformly convergent. Dec 20, 2020 at 20:55
• You can also consider the $L^p$ spaces where we can find a sequence of functions that converge in the $L^p$ norm but don't converge pointwise. Likewise there exists a sequence of functions converging pointwise almost everywhere that does not converge in the $L^p$ norm. So really there is no general connection between pointwise, uniform, and "normal" convergence for a general normed space. Dec 21, 2020 at 2:15