In Analysis I know that a series of functions $$ \{f_n\}_{n=0}^{\infty}$$ converges to $$f$$ if and only if $$\lim\sup\limits_{n\rightarrow\infty}(f -f_n) = 0$$
Now for Fourier series, uniform convergence requires convergence in the infinity/max norm. $$\sum_{n=0}^{\infty}f_n \overset{unif}\rightarrow f$$ if and only if $$ S_N(x) = \sum_{n=0}^{N}f_n$$ $$ \lim\sup_{n\rightarrow{\infty}}\|f-S_N(x)\|_{\infty} $$
But I was told that the limsup in terms of norms is not the same and also that convergence in the infinity norm is not the same as convergence of limsup for the sequence of functions $$\{f_n\}_{n=0}^{\infty}$$
Uniform convergence is stronger because the rate of convergence needs to be the same at all points $x$. The statement $\limsup_{n\rightarrow\infty} |f(x)-f_n(x)|=0$ says $f_n$ converges to $f$ at each point $x$ but does not guarantee uniform convergence. Whereas the infinity norm $\|f-f_n\|_\infty<\epsilon$ implies that $|f(x)-f_n(x)|<\epsilon$ for all $x$ for $n$ large enough.