Let $(a\_n)\_{n=0}^\infty$ be an arbitrary sequence in a real Banach space $X$. Does there exist a smooth function $f: \mathbb {R} \rightarrow X$ such that $f^{(n)}(0)=a\_n$ for $n=0,1,2,\ldots $?



Borel's theorem for Banach's space valued functions
According to *The Convenient Setting of Global Analysis* (Kriegl and Michor), this is due to Wells (1973). The statement given is:



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> **15.4. Borel's Theorem.** [Wells, 1973]. Suppose a Banach space $E$ has $C^\infty\_b$-bump functions. Then every formal power series with coefficients in $L\_{\text{sym}}^n(E;F)$ for another Banach space $F$ is the Taylor-series of a smooth mapping $E \to F$.
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In this case, $E = \mathbb{R}$ so you have $C^\infty\_b$-bump functions.


### References:


* *The Convenient Setting of Global Analysis*, Kriegl and Michor. [MR1471480](http://www.ams.org/mathscinet-getitem?mr=1471480)
* *Differentiable functions on Banach spaces with Lipschitz derivatives*, Wells. [MR370640](http://www.ams.org/mathscinet-getitem?mr=370640)


