Let $U$ be a bounded open set in $\mathbb{R}^n$. Consider $C(U)$, the space of the continuous functions on $U$. I am confused with "uniform convergence" and "$L^{\infty}$-convergence" on $C(U)$. It seems to me that the two convergences are equivalent. If I am wrong, is one of them stronger than the other?
Thank you in advance.
Uniform convergence of say, $f_n$ to $f$, on a set $S$ means that $\sup_{x\in S}|f_n(x)-f(x)|\to 0$ as $n$ goes to infinity.
In the definition of $L^\infty$ norm, there is an idea of almost everywhere. More precisely, if $f$ is a measurable function, then $\lVert f\rVert_{L^\infty}\geqslant |f(x)|$ for almost every $x$. So convergence in $L^\infty$ actually means "uniform convergence on $S\setminus N$ where $N$ has zero-measure".
Now a good exercise is to see how are the uniform norm and the $L^\infty$ norms related when they are applied to continuous functions.
Answer: they are actually the same thing.