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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?

Git Gud
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JackReacher
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  • $f_n(x)$ is a single function (rather, the value of a function $f_n$ at $x$). $(f_n)$ is a sequence of functions. Contrast with a number $n$ and a sequence $(n)$ (more properly written $(n)_{n=1}^\infty$). – David Mitra May 27 '14 at 11:04
  • But without an example by the OP in context, it is quite possible the author uses the notation $f_n(x)$ for a sequence of functions, such as the sequence of power functions $x^n$ without braces or parentheses. It is a common notation, even if it could be called logically incorrect. – KCd May 27 '14 at 11:08
  • For example, say $f_n(x) = \frac{nx^3}{n+x}$ what would be the difference between determining whether $f_n(x) \rightarrow f$ uniformly and say, 'show that the sequence $(f_n)$ converges uniformly? – JackReacher May 27 '14 at 11:14
  • @JackReacher Saying $f_n(x)\to f$ is senseless here. What's being asked is to prove that $(f_n)$ converges uniformly. – Git Gud May 27 '14 at 11:18
  • @GitGud - sorry bad notation on my part. My question is: is there a difference between determining whether a particular $f_n(x)$ converges uniformly (after determining a pointwise convergence) and showing whether $(f_n)$ converges uniformly? – JackReacher May 27 '14 at 11:29
  • I don't understand what you mean with $f_n(x)$ here. – Git Gud May 27 '14 at 11:34
  • For example, as above if $f_n(x) = \frac{nx^3}{n+x}$ what is the difference between the question: show that the sequence $(f_n)$ converges uniformly vs. Determine the pointwise convergence and whether it converges uniformly. – JackReacher May 27 '14 at 11:55
  • "Whether is converges uniformly". What does 'it' refer to? – Git Gud May 27 '14 at 12:37
  • Find the pointwise limit of $f_n(x)$ (function defined as above), and determine whether or not the convergence is uniform. Is this different from: does $(f_n)$ converge uniformly? (if we literally just swapped these two sentences for the same $f_n(x)$) – JackReacher May 27 '14 at 12:42
  • My interpretation is that you're being asked to, given $x\in \mathbb R$, find $x\mapsto \lim \limits_{n\to \infty}(f_n(x))$. Next you're being asked if $(f_n)$ converges uniformly. It doesn't make sense to ask if $f_n(x)$ converges uniformly. Did I answer your question? – Git Gud May 27 '14 at 12:56
  • It may be helpful to quote Ahlfors here, specifically the first footnote in chapter 2 of his "Complex Analysis": Modern students are well aware that $f$ stands for the function and $f(z)$ for a value of the function. However, analysts are traditionally minded and continue to speak of "the function $f(z)$." – Daniel Fischer May 27 '14 at 14:38
  • @JackReacher Do you still need help with this? – Git Gud May 29 '14 at 09:54
  • @GitGud - Thanks for following up. From what I can tell, the two things are asking the same thing. Asking if $f_n(x)$ converges uniformly to some $f(x)$ is the same as determining whether $(f_n)$ converges uniformly (to some $f$) – JackReacher May 29 '14 at 10:24
  • @JackReacher Yes, that's my interpretation of the (badly written) statement of the problem. – Git Gud May 29 '14 at 10:25
  • @JackReacher In view of the discussion in the comments above, perhaps youdidn't ask exactly what you wanted to ask? Do you care to edit your question? – Git Gud Jun 08 '14 at 01:26

1 Answers1

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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.

Git Gud
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