I am reading "Linear Algebra" by Takeshi Saito.

Let $f \in C^\infty(\mathbb{R})$.
Let $T$ be a mapping such that $T: C^\infty(\mathbb{R}) \ni f \mapsto (a_n) = (f^{(n)}(0)) \in \mathbb{R}^\mathbb{N}$.

$C^\infty(\mathbb{R})$ and $\mathbb{R}^\mathbb{N}$ are linear spaces.
$T$ is a linear mapping.

In this book, the author says that $T$ is surjective without a proof.

Why is $T$ surjective?

  • $\begingroup$ I don't know if there's an easy way to see this. Borel's lemma should help in either case. $\endgroup$ – MSDG May 12 '19 at 10:09

The approach of @Monadologie works if you cut off the polynomials early enough. This is the proof presented on Wikipedia for Borel's lemma from a bird's view

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