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I have a (separable) Banach space $E$ and two closed disjoint sets $F$, $G$ in $E$. Now I wish to prove the existence of a $C^2$-function (Fréchet differentiable) $f:E \to \mathbf R$ that is $1$ on $F$ and $0$ on $G$.

Does someone have a reference for this (if it is possible)? If it is not possible, are there additional conditions on the Banach space to make this possible?

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The abstract of this article (JSTOR) looks as if the answer should be expected to be no in general. I can't see more than one page, though, in particular I can't tell whether the examples are separable. – t.b. May 11 '11 at 13:23
Theo, thanks. It seems to fail. See MathSciNet: – Jonas Teuwen May 11 '11 at 13:30
up vote 6 down vote accepted

Have a look at

  • Kriegl, Michor: "The Convenient Setting of Global Analysis",

Chapter III Partitions of Unity.

General Banach spaces don't have bump functions of the sort you seek, for example

14.11 (1): No Fréchet-differentiable bump function exists on C[0,1] and on $l^1$


14.12 (2) If a Banach space and its dual admit $C^2-$ bump functions, then it is isomorphic to a Hilbert space.

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Great, thanks Tim! – Jonas Teuwen May 11 '11 at 13:57
You're welcome! There are also positive results, like theorem 16.11: A separable Banach space has $C^1$-bump functions if E' is separable... – Tim van Beek May 11 '11 at 14:01

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