A residual set of continuous functions has unbounded partial sums at zero

Real continuous periodic functions identify isometrically with $X=\{f\in C([0,1],\mathbb{R}):f(0)=f(1)\}$. A Cauchy sequence has a continuous uniform limit by completeness on the nonempty compact interval. Endpoint equality passes to that limit because evaluation at either endpoint changes by at most the uniform error. Hence $X$ is Banach.

The family $T_N:X\to \mathbb{R}$ consists of bounded linear maps, with unbounded operator norms. The bounded-norm alternative in the dichotomy is therefore excluded; its other alternative says precisely that $\mathcal{U}$ is dense and $G_{\delta }$ in $X$.

Explicitly, $\mathcal{U}={\bigcap }_{r=1}^{\mathrm{\infty }}{\bigcup }_{N=0}^{\mathrm{\infty }}\{f\in X:\mathrm{\mid }{T}_{N}f\mathrm{\mid }>r\}$. Each inner set is open by continuity of $T_N$, and the displayed membership condition is exactly unboundedness of the sequence of finite values. Thus the topology in this conclusion is topology on a space of functions; it asserts no full-measure set of points of the torus for any fixed function.