# Weaker version of the Borel lemma for vector-valued functions

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?

• @PietroMajer Yes, you're right. Thanks! Dec 28 '20 at 11:23