Understanding the definition of Uniform Absolute Convergence The definition of uniform absolute convergence given here: https://proofwiki.org/wiki/Definition:Uniform_Absolute_Convergence states that $\sum_{n=1}^\infty f_n$ converges uniformly absolutely if $\sum_{n=1}^\infty \|f_n\|$ converges uniformly.
I am a bit confused here by the taking of the norm of $f_n$ in the definition. The maps $f_n: S \rightarrow V$ map into the vector space, but the function itself is not in the vector space. So how come we can take the norm of the function? I.e. what does it mean to take the norm here?
Also the definition on wikipedia on the other hand is $\sum_{n=1}^\infty f_n(x)$ converges uniformly absolutely if $\sum_{n=1}^\infty |f_n(x)|$ is uniformly convergent. This makes more sense to me, as in this case, the definition just says that the series of the modulus of functions converges uniformly. I am not sure how to unify the two definitions.
My notes in real analysis introduce the same notation as the proofwiki definition, that is that we have uniform absolute convergence if $\sum_{n=1}^\infty \|f_n\|_{C(i)} < \infty$. Unfortunately, in the real analysis course we were told this was just a useful piece of notation that meant taking the supremum. The details were meant to be filled in during later courses in analysis. However now in Fourier analysis we are using the notation in more contexts, but I haven't actually been able to pick up on what it really means.
 A: In finite dimensions, all norms on a vector space are equivalent. Thus we tend to stick with the well-understood Euclidean norm on $\Bbb R^n$, occasionally pulling out the sup or taxicab norms when computationally convenient.
But most function spaces are infinite-dimensional, which means norms finally have enough freedom to disagree with each other. And the thing is, there are several different norms that are useful. And when as here, we are talking about several possible spaces, each with its own norm, it becomes confusing which norm is being referred to.
There are only two notations for norms that are so widely used that an author can use them without explaining. These are:

*

*$\|\cdot\|_V$ is the norm on the space $V$. You can only use it when in context you've already defined a particular norm on $V$. Presumably, your professor had already mentioned they were using the sumpremum norm on $C(i)$ (maybe $C(I)$, where $I = [0,1]$?).

*$\|\cdot\|_p$, where $p$ is any real number $\ge 1$, or $p = \infty$. For $p < \infty$, these refer to the function norms $$\|f\|_p := \left[\int_M |f|^p\,d\mu\right]^{1/p}$$ defined for  functions $f$ on some measure space $M$ for which $|f|^p$ is integrable. $\|f\|_\infty = \sup_{x \in M} |f(x)|$, and is given that notation because $\lim_{p \to \infty} \|f\|_p = \|f\|_\infty$
In specialized fields of mathematics, there are some other notations that are well-known in that field, and may be used when speaking to those well-versed in the field. But when speaking to a wider mathematically inclined audience, the notations above are the only ones that can safely be used without explanation. And of course even then, there will always be a few novices who might need an introduction.

The tip-off that Daniel Fischer is correct about the intent here is the definition specifies that $\sum_{n=1}^\infty \|f_n\|$ must converge uniformly.  If $\|f_n\|$ meant the supremum norm of $f_n$, then the series would be a sum of real numbers, and talking about "uniform" convergence makes no sense. Uniform with respect to what? In order to talk about uniform convergence, $\sum \|f_n\|$ must be a sum of functions. The interpretation that makes $\|f_n\|$ a function is that it means the function $$\|f_n\| : S \to \Bbb R : x \mapsto \|f_n(x)\|$$
