# smooth functions on closed intervals with values in infinite-dimensional spaces

There are three ways to define when a ($$\mathbb{R}$$-valued) function on a closed interval is smooth:

1. $$f$$ can be extended to a smooth function on $$(a - \epsilon, b + \epsilon)$$ for some $$\epsilon > 0$$
2. $$f \in \bigcap_{k \geq 0} C^k([a, b])$$. Here $$f \in C^1([a, b])$$ when the limit of the differences quotients of $$f$$ is exists at every point and defines a continuous function on $$[a, b]$$ and $$f \in C^k([a, b])$$ if $$f' \in C^{k-1}([a, b])$$. $$\label{Ck}$$
3. $$f$$ is smooth on $$(a, b)$$ and all of its derivatives extend continuously to $$[a, b]$$

Obviously $$(1) \Rightarrow (2) \Rightarrow (3)$$ and to show $$(3) \Rightarrow (1)$$ one can use that it's possible to construct smooth functions with arbitrary Taylor series at some point, as is for example done in [1, Lemma A.1.3].

I'm interested in the case where the function takes values in some kind of topological vector space $$E$$. A derivative of $$f$$ can still be defined as the limit of difference quotients.

Q: For what kind of TVSs do some of these three definitions coincide?

The cited constructions uses an absolutely and uniformly convergent series to define a smooth extension of function satisfying $$(3)$$, so I think it still works whenever $$E$$ is a Banach space.
I'm most interested in the case where $$E$$ is a Fréchet space.

Reference:

1. McDuff and Salamon, Introduction to Symplectic Topology, 3e, Oxford Graduate Texts in Mathematics
Here is the relevant Lemma

I think that the answer to your question follows from the usual tensor representation of Frechet valued smooth functions: $$C^\infty(\mathbb R,E)=C^\infty(\mathbb R) \hat\otimes_\pi E$$ and $$C^\infty([a,b],E)=C^\infty([a,b]) \hat\otimes_\pi E$$ (I did not check for references, this is probably already in Grothendieck's thesis). In the scalar case, the restriction map $$r:C^\infty(\mathbb R)\to C^\infty([a,b])$$ is surjective (even more, by a theorem of Mityagin and independently Seeley, it has a continuous linear right inverse) and the $$\pi$$-tensor product of Frechet spaces respects quotients and therefore the Frechet valued restriction mapping $$R=r\otimes id_E$$ is also surjective.
(3) $$\Rightarrow$$ (2) and hence (2) $$\Leftrightarrow$$ (3) whenever $$E$$ is any Hausdorff locally convex space. This is a simple consequence of the vector valued mean value theorem that in turn by Hahn−Banach follows easily from the result for real valued functions on a closed real interval.