nLab Boman's theorem

Context

Differential geometry

differential geometry

synthetic differential geometry

Applications

In the paper Differentiability of a function and of its compositions with functions of one variable, Jan Boman proves the following theorem:

Theorem

Let $f$ be a function from ${ℝ}^{d}$ to $ℝ$, and assume that the composed function $f\circ u$ belongs to ${C}^{\infty }\left(ℝ,ℝ\right)$ for every $u\in {C}^{\infty }\left(ℝ,{ℝ}^{d}\right)$. Then $f\in {C}^{\infty }\left({ℝ}^{d},ℝ\right)$.

The theorem is quoted with a proof in The convenient setting of global analysis by Kriegl and Michor (theorem 3.4).

References

• Jan Boman, Differentiability of a function and of its compositions with functions of one variable, Math. Scand. 20 1967 249–268, MR237728 pdf

• Andreas Kriegl, Peter W. Michor, The convenient setting of global analysis, Math. Surveys and Monographs 53, Amer. Math. Soc. 1997. x+618 pp. ISBN: 0-8218-0780-3 html MR1471480

Revised on November 10, 2012 02:06:15 by Zoran Škoda (193.55.36.32)