A description/proof of the inverse theorem for Euclidean-normed spaces map
Topics
About: normed space
About: map
The table of contents of this article
Starting Context
- The reader knows a definition of Euclidean-normed space.
- The reader knows a definition of map.
- The reader knows a definition of derivative of normed spaces map.
- The reader admits the contraction mapping principle.
- The reader admits the proposition that derivative of \(C^1\), Euclidean-normed spaces map is the Jacobian.
- The reader admits the proposition that residue of derivative of normed-spaces map is differentiable at point of the 2nd argument if the original map is differentiable at the corresponding point.
Target Context
- The reader will have a description and a proof of the inverse theorem for Euclidean-normed spaces map.
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 spaces, \(\mathbb{R}^d\) and \(\mathbb{R}^d\), and any neighborhood, \(U_{v_{11}}\), of point, \(v_{11} \in \mathbb{R}^d\), any map, \(f: U_{v_{11}} \rightarrow \mathbb{R}^d\), such that it is \(C^k\) (where \(1 \le k\)) in \(U_{v_{11}}\) and its derivative at \(v_{11}\), \(Df (v_{11})\), is invertible, is invertible in a neighborhood of \(f (v_{11}) = v_{21})\), and the inverse map is \(C^k\).
2: Proof
$$f (v_{11} + v_{12}) = f (v_{11}) + (Df (v_{11})) (v_{12}) + r (v_{11}, v_{12}) = v_{21} + v_{22},$$ which has to be solved for \(v_{12}\) by \(v_{22}\) as \(v_{12} = g (v_{22})\). $$v_{12} = (Df (v_{11}))^{-1} (v_{22} - r (v_{11}, v_{12})) := T_{v_{22}} (v_{12}).$$ Define the set, \(S_{v_{22}} = \{v_1| v_{11} + v_1 \in U_{v_{11}} \text{ and } \Vert v_1 - ( (Df (v_{11}))^{-1}) (v_{22})\Vert \le M_{v_{22}}\}\) where \(M_{v_{22}} := sup_{\Vert v_1\Vert \le 2\Vert ( (Df (v_{11}))^{-1}) (v_{22})\Vert} \Vert ( (Df (v_{11}))^{-1}) (r (v_{11}, v_1))\Vert\). The map, \(T_{v_{22}}\), with the domain as \(S_{v_{22}}\), is in fact \(T_{v_{22}}: S_{v_{22}} \rightarrow S_{v_{22}}\) if \(\Vert ( (Df (v_{11}))^{-1}) (v_{22})\Vert\) is small enough by the following reason: as \(\Vert v_1 - ( (Df (v_{11}))^{-1}) (v_{22})\Vert \le M_{v_{22}}\), \(\Vert v_1\Vert - \Vert ( (Df (v_{11}))^{-1}) (v_{22})\Vert \le M_{v_{22}}\), so, \(\Vert v_1 \Vert \le \Vert ( (Df (v_{11}))^{-1}) (v_{22})\Vert + M_{v_{22}}\), but for any \(0 \lt \varepsilon\), there is a \(\delta\) such that for \(\{\forall v_1| \Vert v_1 \Vert \lt \delta\}\), \(\frac{\Vert r (v_{11}, v_1) \Vert}{\Vert v_1 \Vert} \lt \varepsilon\), so, \(\Vert ( (Df (v_{11}))^{-1}) (r (v_{11}, v_1)) \Vert \le \Vert ( (Df (v_{11}))^{-1}) \Vert \Vert (r (v_{11}, v_1)) \Vert \lt \Vert ( (Df (v_{11}))^{-1}) \Vert \Vert v_1 \Vert \varepsilon\), so, if \(2 \Vert ( (Df (v_{11}))^{-1}) (v_{22}) \Vert \lt \delta\), \(M_{v_{22}} \le \Vert ( (Df (v_{11}))^{-1}) \Vert 2 \Vert ( (Df (v_{11}))^{-1}) (v_{22}) \Vert \varepsilon\), then, take \(\varepsilon = 2^{-1} \Vert ( (Df (v_{11}))^{-1}) \Vert ^{-1}\), then, \(M_{v_{22}} \le \Vert ( (Df (v_{11}))^{-1}) (v_{22}) \Vert\), so, \(\Vert v_1 \Vert \le \Vert ( (Df (v_{11}))^{-1}) (v_{22}) \Vert + M_{v_{22}} \le 2 \Vert ( (Df (v_{11}))^{-1}) (v_{22}) \Vert\), then, \(\Vert T_{v_{22}} (v_1) - ( (Df (v_{11}))^{-1}) (v_{22}) \Vert = \Vert ( (Df (v_{11}))^{-1}) (r (v_{11}, v_1)) \Vert \le M_{v_{22}}\). \(T_{v_{22}}\) with the domain is a contraction with norm of subtraction as distance for a small enough \(v_{22}\), because \(\Vert T_{v_{22}} (v_{121}) - T_{v_{22}} (v_{122}) \Vert = \Vert (Df (v_{11}))^{-1} (r (v_{11}, v_{122}) - r(v_{11}, v_{121})) \Vert \le \Vert (Df (v_{11}))^{-1} \Vert \Vert r (v_{11}, v_{122}) - r(v_{11}, v_{121}) \Vert\), but r is differentiable with respect to the 2nd argument and the Jacobian is continuous because it is expressed by some Jacobians of f, which are continuous, so, by the mean value theorem, it is \(\Vert (Df (v_{11}))^{-1} \Vert \Vert Dr (v_{11}, v_{123}) \Vert \Vert v_{122} - v_{121} \Vert\), but as \(Dr (v_{11}, 0) = 0\) and continuous, if \(v_{22}\) is small enough, \(v_{123}\) is small enough, and \(\Vert (Df (v_{11}))^{-1} \Vert \Vert Dr (v_{11}, v_{123}) \Vert \lt 1\). So, by the contraction mapping principle, there is the unique fixed point, \(v_{12}\), such that \(T_{v_{22}} (v_{12}) = v_{12}\), which is the desired value. As \(v_{12} \in S_{v_{12}}\), \(\Vert v_{12} - ( (Df (v_{11}))^{-1}) (v_{22}) \Vert \le M_{v_{22}}\), but \(lim_{\Vert v_{22} \Vert \rightarrow 0} \frac{M_{v_{22}}}{\Vert v_{22} \Vert} = 0\) (which comes from the definition of \(M_{v_{22}}\) and \(lim_{\Vert v_1 \Vert \rightarrow 0} \frac{\Vert r (v_{11}, v_1) \Vert}{\Vert v_1\Vert}\)), which exactly means that \(v_{12} = g (v_{22})\) is differentiable at 0 with the derivative as \((Df (v_{11}))^{-1}\), which means that \(f^{-1}\) is differentiable with the same derivative because \(f^{-1} (v_{21} + v_{22}) = v_{11} + g (v_{22}) = f^{-1} (v_{21}) + (Dg (0)) (v_{22}) + r (0, v_{22})\). \(D(f^{-1}) = (Df (v_{11}))^{-1} = (Df (f^{-1} (v_{21})))^{-1}\), is continuous with respect to \(v_{21}\), because \(f^{-1}\) is continuous being differentiable, the Jacobian is continuous with respect to \(v_{11}\), having the reverse matrix preserves the continuousness, and the compound of such continuous maps is continuous.
Now that the theorem has been proved to be true for \(C^1\), suppose that the theorem is true for \(C^{k - 1}\) and f is \(C^k\). As \(D (f^{-1}) = [\frac{\partial f}{\partial v_{1}}]^{-1}\), the right hand side is \(C^{k - 1}\) with respect to \(v_1\) because \([\frac{\partial f}{\partial v_{1}}]\) is \(C^{k - 1}\) as f is \(C^k\) and taking the reverse matrix preserves the \(C^{k - 1}\)-ness, and as \(v_1 = f^{-1} (v_2)\) is \(C^{k - 1}\) by the theorem for \(C^{k -1}\), as the compound map, the right hand side is \(C^{k - 1}\) with respect to \(v_2\), also the left hand side is so, which means that \(f^{-1}\) is \(C^k\).
3: Note
The area of possible \(v_{22}\)s is a concern, and is determined as this, although this is not very straightforward: 1st, take \(\varepsilon = 2^{-1} \Vert ( (Df (v_{11}))^{-1}) \Vert ^{-1}\) and take a \(\delta\) such that for \(\{\forall v_1| \Vert v_1 \Vert \lt \delta\}\), \(\frac{\Vert r (v_{11}, v_1) \Vert}{\Vert v_1 \Vert} \lt \varepsilon\) and take \(v_{22}\) as \(2 \Vert ( (Df (v_{11}))^{-1}) (v_{22}) \Vert \lt \delta\); 2nd, belittle \(v_{22}\) as \(\Vert (Df (v_{11}))^{-1} \Vert \Vert Dr (v_{11}, v_1) \Vert \lt 1\) for \(\{\forall v_1| \Vert v_1 - ( (Df (v_{11}))^{-1}) (v_{22}) \Vert \le sup_{\Vert v_1 \Vert \le 2 \Vert ( (Df (v_{11}))^{-1}) (v_{22}) \Vert} \Vert ( (Df (v_{11}))^{-1}) (r (v_{11}, v_1)) \Vert\}\).
