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Hadamard's lemma : ウィキペディア英語版
Hadamard's lemma
In mathematics, Hadamard's lemma, named after Jacques Hadamard, is essentially a first-order form of Taylor's theorem, in which we can express a smooth, real-valued function exactly in a convenient manner.
==Statement==
Let ƒ be a smooth, real-valued function defined on an open, star-convex neighborhood ''U'' of a point ''a'' in ''n''-dimensional Euclidean space. Then ƒ(''x'') can be expressed, for all ''x'' in ''U'', in the form:
:f(x)=f(a)+\sum_^n (x_i-a_i) g_i(x),
where each ''gi'' is a smooth function on ''U'', ''a'' = (''a''1,...,''a''''n''), and ''x'' = (''x''1,...,''x''''n'').

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