What is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$?

Question

What is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$? I am reading my textbook on analysis, and it seems to use 'sequence of functions' to describe both $(f_n)$ and $f_n(x)$. Could someone help me with I suppose an intuitive explanation of the difference?

Answer 1

A sequence is any map whose domain in the natural numbers, that is, it is a function $x\colon \mathbb N\to \text{Somewhere}$. The name of the sequence is $x$ and the image of each element $n\in \mathbb N$ is $x(n)$ but often abbreviated as $x_n$. It is common to denote $x$ by $(x_n)_{n\in \mathbb N}$.

In this case you have $(f_n)_{n\in \mathbb N}$, where presumably $f_n$ are functions whose domain and image are subsets of $\mathbb R$. If $x\in \mathbb R$, the notation $(f_n(x))_{n\in \mathbb N}$ is not a sequence of functions, it's a regular sequence where $x$ is acting out as a parameter. The correct notation is $(f_n)_{n\in \mathbb N}$.

It should be noted that the notation $(f_n)_{n\in \mathbb N}$ yields some ambiguity because $f$ is denoting two different things here. One of them is the sequence whose image of an element $n\in \mathbb N$ is determined by $f(n)=f_n$, it is a sequence. The other one is the function $x\mapsto \lim \limits_{n\to \infty}(f_n(x))$, the pointwise convergence function. In this context the first meaning of $f$ given is usually abandoned in favor of the latter.

Could someone help me with I suppose an intuitive explanation of the difference?

Intuitively, for some people, there is not difference. The authors mean the same with $(f_n(x))_{n\in \mathbb N}$ as they do with $(f_n)_{n\in \mathbb N}$. The use of the (actually inaccurate) $(f_n(x))_{n\in \mathbb N}$ is to remind the reader that $f_n$ are 'functions of $x$' or functions of one real variable.