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Summary: The Implicit Function Theorem. Suppose
(1) X, Y and Z are Banach spaces;
(2) C is an open subset of X × Y ,
f : C Z
and f is continuously differentiable on C;
(3) (a, b) C and
Y v f(a, b)(0, v)
is a Banach space isomorphism from Y onto Z;
Then there are an open subset U of X such that a U; an open subset W of Z such that f(a, b) W;
an open subset V of X × Y such that (a, b) V C; and g such that
(4) g : U × W Y and g is continuously differentiable on U × W;
(5)
(x, y) V and z = f(x, y) (x, z) U × W and y = g(x, z).
Remark. Note that C = {(x, g(x, z)) : (x, z) U × W}.
Proof. Let F(x, y) = (x, f(x, y)) for x C. By the Corollary to the Inverse Function Theorem, the Chain
Rule and the smoothness of inversion we obtain an open subset D of X such that (a, b) D C and
(6) F[D] is an open subset of Y × Z;
(7) F|D is univalent;
(8) (F|D)-1
is continously differentiable.
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