Abstract In the hypthesis that f is analytic, Smale gave an estimation of the size of the radius of the ball in which the inverse function exists, by using the criterion α. In this paper we not only make this estimation more precise, but also weaken the hypothesis to the second continuous differentiable. Keywords Banach space, The inverse function theorem, α-criterion 1991 MR Subject Classification 58C15, 46E15 Chinese Library Classification O177.91
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