Exact meaning of $\sup_n\|\sum_{k=0}^ng_k\|_\infty<\infty$

Question

I dont have totally clear the meaning of the expression

$$\sup_{n\in\Bbb N}\left\|\sum_{k=0}^ng_k\right\|_\infty<\infty$$

for a series of functions $g_k$ valued in $\Bbb R$ or $\Bbb C$. In particular I dont know if the supremum over the naturals have the infinity included as a "limit point" or so. The difference is very important:

• if infinity is a "limit point" of $\Bbb N$ then the above expression doesnt necessarily imply that the series eventually decreases or remain constant.

• if infinity is not a limit point of $\Bbb N$ then the above expression imply that the series eventually decreases or stay constant.

Unfortunately the context where I get this (an exercise) dont show clearly what is the exact meaning. I would like to assume the second, what would simplify the exercise dramatically, but Im not sure. Some help will be appreciated.

Answer 1

I'm not quite sure what you mean by including infinity as a limit point of $\mathbb{N}$ but I will offer you my interpretation of what the expression is telling you.

It seems to me that $\{g_k\}$ is a sequence of functions, and essentially here we've defined another sequence of functions, say $f_n$, by $$f_n(x)=\sum_{k=1}^{n}g_k(x).$$ So the expression merely says that $\sup_{n\in\mathbb{N}}||f_n||_\infty <\infty$. That is, $f_n$ is uniformly bounded by some finite $M<\infty$. Then, in particular, for all $x\in\mathbb{R}$ or $\mathbb{C}$ or whatever the underlying domain is, and for all $n$ we must have $$|f_n(x)|=|g_1(x)+...+g_n(x)|\leq M.$$ As for the individual $g_k$, it follows that $|g_k(x)|=|f_{k}(x)-f_{k-1}(x)|\leq |f_{k}(x)|+|f_{k-1}(x)|\leq 2M<\infty$. With all that said, I don't think you can say much about the increasing/decreasing nature of $f_n$.

Answer 2

This is just the supremum of a countable set of real numbers, right? Written differently, it is $$\sup\left\{ \|S_1\|_{\infty}, \|S_2\|_{\infty}, \|S_3\|_{\infty}, \ldots \right\}$$ where $S_n$ is the $n^{\textrm{th}}$ partial sum: $S_n=\sum\limits_{k=0}^n g_k$. And the statement is just that this supremumum is finite.