... inequality,1
The triangle inequality, which is one of the defining properties of a norm, states that

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... inequality,2
The reverse triangle inequality, which can be proved from the triangle inequality, states that

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... integrand.3
A digression: Equation (4) is commonly used when one wants to use a Mean Value Theorem (MVT). For a function of one variable, the MVT (a special case of Taylor's theorem) states that if f is sufficiently smooth, then there exists such that

f(b)=f(a)+f'(c)(b-a).

However, the MVT does not hold for vector-valued functions, because the number c is typically different for each component Fi. Equation (4) is usually an adequate substitute.
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Mark S. Gockenbach
2003-01-23