As we know, for $1<p<\infty$, the Fourier series of $f\in L^{p}(T)$ converges to $f$ in $L^{p}$norm. But is there any results concerning the convergence of Fourier series in $L^{\infty}$norm? Since $L^{\infty}(T)$ is not separable, the trigonometric system fails to form a Schauder basis of $L^{\infty}(T)$, this implies that the Fourier series of $L^{\infty}(T)$functions fails to converge in $L^{\infty}$norm. But does the Fourier series of $f$ converge in $L^{\infty}$norm for every $f\in C(T)$?

closed as too localized by Harald HancheOlsen, Willie Wong, Andres Caicedo, Mark Meckes, Yemon Choi Apr 3 '11 at 19:00
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If Fourier series of continuous functions would converge in $L^\infty$, then, by the Uniform Boundedness Principle, the operator norms in $C(\mathbb{T})$ of the partial Fourier series operators $S_Nf(t):=\sum_{n=N}^N\hat{f}(n)e^{int}$ would be uniformly bounded. You can find, for example in Katznelson book, a proof of the fact that such norms diverge logarithmically. 


I think you can find the answer in P191 "Classical Fourier Analysis",second edition, Loukas Grafakos 

