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
- The reader knows a definition of normed Euclidean \(C^\infty\) manifold.
- The reader knows a definition of derivative of normed spaces map.
- The reader knows a definition of closure of subset of topological space.
- The reader admits the mean value theorem for any differentiable function from any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold.
- The reader admits the Heine-Borel theorem: any subset of any Euclidean topological space is compact if and only if it is closed and bounded.
- The reader admits the proposition that the image of any continuous map from any compact topological space to any Euclidean topological space has the minimum and the maximum.
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
There is a list of definitions discussed so far in this site.
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, \(\Vert f (p_2) - f (p_1)\Vert \leq L \Vert p_2 - p_1\Vert\).
