2022-09-25

353: C^1 Map from Open Set on Euclidean Normed C^\infty Manifold to Euclidean Normed C^\infty Manifold Locally Satisfies Lipschitz Condition

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A description/proof of that \(C^1\) map from open set on Euclidean normed \(C^\infty\) manifold to Euclidean normed \(C^\infty\) manifold locally satisfies Lipschitz condition

Topics


About: Euclidean normed \(C^\infty\) manifold
About: map

The table of contents of this article


Starting Context



Target Context


  • The reader will have a description and a proof of the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.

Orientation


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There is a list of propositions discussed so far in this site.


Main Body


1: Description


For any Euclidean normed \(C^\infty\) manifold, \(\mathbb{R}^{d_1}\), any open set, \(U \subset \mathbb{R}^{d_1}\), and any Euclidean normed \(C^\infty\) manifold, \(\mathbb{R}^{d_2}\), any \(C^1\) map, \(U \rightarrow \mathbb{R}^{d_2}\), satisfies the Lipschitz condition on any convex open set, \(U' \subseteq U\), whose closure is bounded and is contained in \(U\), which is, there is a constant, L, such that for any points, \(p_1, p_2 \in U'\), \(\Vert f (p_2) - f (p_1)\Vert \leq L \Vert p_2 - p_1\Vert\).


2: Proof


On \(U'\), by the mean value theorem for any differentiable function from any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold, \(\Vert f (p_2) - f (p_1)\Vert \leq \Vert Df (p_3)\Vert \Vert p_2 - p_1\Vert\) where \(p_3\) is a point on the line segment from \(p_1\) to \(p_2\) and \(\Vert Df (p_3)\Vert\) is the matrix norm (induced by vector norms) of the Jacobian. As each matrix element is continuous, it has the minimum and the maximum on the closure of \(U'\), because the bounded closet set is compact by the Heine-Borel theorem, and the image of any continuous map from any compact topological space to the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum, so, on \(U'\), each matrix element has a finite infimum and a finite supremum, so, the matrix norm has a finite supremum, L. So, \(\Vert f (p_2) - f (p_1)\Vert \leq L \Vert p_2 - p_1\Vert\).


References


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