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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 :)

Philipp
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I believe when you say "normal" convergence that it actually is the same thing as pointwise convergence. What would it mean for a sequence of functions to converge to a function in a function space? Functions are defined by acting on elements of a given set, so the only way to check convergence would be to check how they act on their domain, but this is just how they act for each point, hence pointwise convergence.

So really there only is a distinction between pointwise and uniform convergence. This difference can best be summarized by the following: pointwise convergence is concerned with a single point at a time. Uniform convergence is concerned with ALL points in the domain at the SAME time. This is important because while a sequence of functions may converge pointwise, if it is converging at different "rates" at each point in may not converge uniformly.

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 $f_n(x) <\varepsilon$

Highlander13
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  • 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. – Philipp Dec 20 '20 at 12:44
  • I supported my question with an example. Hopefully it helps :) – Philipp Dec 20 '20 at 12:51
  • Okay I think I am understanding better, my apologies for misinterpreting the question! So you are basically asking that we can have numerous types of convergence of functions that may or may not be connected to pointwise and uniform convergence? And the way to distinguish is to see what space and norm we are using? Is this what you are referring to? – Highlander13 Dec 20 '20 at 19:30
  • 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. – Philipp Dec 20 '20 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. – Highlander13 Dec 21 '20 at 02:15