I want to show that all absolute convergent series in $L^\infty$ with $||.||_\infty$ converges, directly using definition. But I am confused on what exactly do I need to show. My goal is to use this to show that $L^\infty$ is complete.
Definitions in the book I'm reading,
$X$ be a normed vector space, a series in $X$, $\sum x_n$ is said to converge to $x$ if $\sum^N x_n\to x$ as $N\to \infty$ and it is absolutely convergent if $\sum ||x_n||<\infty$
If I supposed that $\{f_n\}$ is a sequence in $L^\infty$ and the series $\sum ||f_n||_\infty<\infty$, do I need to show that for all $\epsilon>0$ there exists $K$ such that $||\sum^N f_n -f||_\infty<\epsilon$ for all $N>K$?
It would be greatly appreciated, if you could provide some insight and logic of how to approach this. Many thanks!
Given $j$ there exists $N$ such that $ \|\sum\limits_{k=n}^{m}f_n\|<\frac 1 j$ whenever $m>n>N$. This implies that $ |\sum\limits_{k=n}^{m}f_n(x)|<\frac 1 j$ almost everywhere whenever $m>n>N$. [Note that there are only a coutable number of null sets involved here and their union is a null set]. Thus, for almost all $x$, $(\sum\limits_{k=1}^{m}f_k(x))$ is a Cauchy sequence of real numbers and $f(x)=\lim \sum\limits_{k=1}^{m}f_n(x)$ exists. To finish the proof observe that $ |\sum\limits_{k=1}^{m}f_n(x)-f(x)|\leq\sum\limits_{k=m+1}^{\infty}\|f_n(x)\|\leq \frac 1 j$ almost everywhere whenever $m>N$. Thus, $ \|\sum\limits_{k=1}^{m}f_n(x)-f(x)\|\leq \frac 1 j$ whenever $m>N$ proving that $\sum_n f_n=f$ in the norm of $L^{\infty}$.