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Boman's theorem

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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∘u belongs to C ∞(ℝ,ℝ) for every u∈C ∞(ℝ,ℝ d). Then f∈C ∞(ℝ d,ℝ).

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)
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