Proposition 5

Tag 084H, permanent. Cite it as [Bourbaki, Tag 084H].

Functions of a Real Variable, chapter I, § 2. THE MEAN VALUE THEOREM

Let $f$ be a vector function of a complex variable, defined, continuous and differentiable on a convex open set $A \subset \mathbf{C}$ , taking values in a normed space over the field $\mathbf{C}$ ; then, no matter what the points $x_0, x$ and $y$ in $A$ , one has

\|f(y) - f(x) - f'_d(x_0)(y - x)\| \leq |y - x| \sup_{z \in A} \|f'(z) - f'(x_0)\|.

(10)

It suffices to apply th. 2 to the function

g(t) = f(x + t(y - x)) - f'(x_0)(y - x)t

on the interval

[0, 1]

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

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Cites

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