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 $C^{\mathrm{\infty }}$ manifold to Euclidean normed $C^{\mathrm{\infty }}$ manifold locally satisfies Lipschitz condition

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Topics

About: Euclidean normed $C^{\mathrm{\infty }}$ manifold
About: map

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of normed Euclidean $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ manifold to any Euclidean normed $C^{\mathrm{\infty }}$ 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.

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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 $C^{\mathrm{\infty }}$ manifold to any Euclidean normed $C^{\mathrm{\infty }}$ manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.

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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.

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Main Body

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1: Description

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

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2: Proof

On $U^{\prime }$, by the mean value theorem for any differentiable function from any Euclidean normed $C^{\mathrm{\infty }}$ manifold to any Euclidean normed $C^{\mathrm{\infty }}$ manifold, $\Vert f(p_2)-f(p_1)\Vert \le \Vert Df(p_3)\Vert \Vert p_2-p_1\Vert$ 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 $U^{\prime }$, 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 $U^{\prime }$, each matrix element has a finite infimum and a finite supremum, so, the matrix norm has a finite supremum, L. So, $\Vert f(p_2)-f(p_1)\Vert \le L\Vert p_2-p_1\Vert$.

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References

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