= Solution
The real space $C(\mathbb T)$ is a <Banach space> in the <uniform norm>: a uniformly <Cauchy sequence> has a uniform pointwise limit by <completeness> of $\mathbb R$, and that <uniform limit> is <continuous>. Each functional $L_N$ from the preceding part is bounded, but their norms are unbounded.
For integers $m\geq1$, define
$$
E_m=\{f\in C(\mathbb T):|L_Nf|\leq m\text{ for every }N\}.
$$
Each $E_m$ is <closed>, being an intersection of inverse images of <closed> intervals under <continuous> functionals. It has empty interior. Otherwise some ball $B(f_0,r)$ would be contained in it. For every $h$ of <norm> at most one, both $f_0$ and $f_0+(r/2)h$ would belong to that ball, so
$$
\frac r2|L_Nh|\leq |L_N(f_0+(r/2)h)|+|L_Nf_0|\leq2m.
$$
Taking the <supremum> over $h$ would give $\|L_N\|\leq4m/r$ for all $N$, contradicting the preceding part.
Thus the $E_m$ are <nowhere dense>. The <Baire category theorem> says their union cannot cover $C(\mathbb T)$. A function outside that union satisfies
$$
\boxed{\sup_N|S_N(f,0)|=\infty,}
$$
so its <Fourier series> diverges at zero. In fact these functions form a <dense> $G_\delta$ subset of $C(\mathbb T)$. This proves the relevant <Uniform boundedness principle> argument directly and establishes <generic unbounded Fourier sums at a fixed point>.
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