In '*Kriegl, Michor - A convenient setting for global infintite-dimensional analysis*', they say that for an element $x$ in a convenient (i.e. Mackey-complete locally convex) space $X$, a bounded derivation $\partial\_x: C^\infty(X) \to \mathbb{R}$ induces a bounded derivation $\partial\_x: C^\infty(U) \to \mathbb{R}$ for $x \in U \subseteq X$ (28.1. - *operational tangent vectors*). In finite dimensions this is clear, since derivatives only depend on the germ, but in the infinite-dimensional case not every derivation is given by a directional derivative, right?



Restriction of derivations on $C^\infty(X)$
In 28.1 of your reference an operational tangent vector is defined more carefully than what you write: On the ring $C^\infty(E\supseteq \{x\},\mathbb R)$ of germs at $x$. This induces a point derivation at $x$ on $C^\infty(U,\mathbb R)$ for each $c^\infty$=set containing $x$. To prove the other direction you need assumptions like smoothly paracompact or smoothly normal for the convenient vector space.


