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Im Bier, Beweis

30 November, 2011

(though wine may be needed for truth)

Worked out last night over a quiet pint:

\begin{aligned} & \quad (3x^2-2x^3)(3y^2-2y^3)-(3z^2-2z^3) \\  & =  6xy(x^2-x)(y^2-y) + (xy-z)^2(3-2xy-4z)+6(xy-z)(z-z^2) \end{aligned}

Consequently: if x, y and z lie in a commutative Banach algebra, and x^2-x, y^2-y, z^2-z and xy-z all have norm O(\varepsilon), then the expression on the left hand side has norm at most O(\varepsilon^2).

[Title corrected 2011-12-08]

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