Borel's lemma for Frechét-spaces $V$ says:
(i) For every $(v_j)_{j \in \mathbb{N}} \in V^\mathbb{N}$ there exists a smooth $f: \mathbb{R} \to V$ such that $$f^{(j)}(0) = v_j.$$
For general locally convex spaces this theorem is false, see e.g. Borel Lemma for vector-valued functions. One then could try to restate the "lemma" in the following weaker way:
(ii) For every $(v_j)_{j \in \mathbb{N}} \in V^\mathbb{N}$ there exists an $f: \mathbb{R} \to V$ such that for every $N \in \mathbb{N}$ $$f(t) = \sum_{j=0}^N \frac{t^j}{j!} v_j + o(|t^N|).$$
This weaker notion of the derivative is called the Peano derivative here: Taylor $k$-differentiability of a real function at a point. I couldn't find anything regarding spaces having property (ii) in the literature.
So my question: Is there a space $V$ such that (ii) holds but (i) doesn't?
