Consider the linear space $\sum_{\mathbb{R}} \mathbb{R}$. As in the Frolicher-Kriegl-Michor view, we make this into a Frolicher space as follows.
1. Equip it with the locally convex topology of the colimit. Specifically, it is given the finest locally convex topology so that all of the inclusions of finite summands are continuous. What is important is that a linear functions $\phi \colon \sum_{\mathbb{R}} \mathbb{R} \to \mathbb{R}$ is continuous if and only if its restriction to any finite summand is continuous. Thus all linear functionals $\sum_{\mathbb{R}} \mathbb{R} \to \mathbb{R}$ are continuous. Technical note: with this topology, it is complete.