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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?

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    $\begingroup$ @PietroMajer Yes, you're right. Thanks! $\endgroup$ Dec 28 '20 at 11:23

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