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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 $m\ge 1$, defineEach $E_m$ 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, soTaking the supremum over $h$ would give $\Vert L_N\Vert \le 4m/r$ 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 satisfiesso 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.