1642: For \(C^\infty\) Manifold with Boundary and \(C^\infty\) Vectors Field over Manifold with Boundary, Integral Curves into Interior of Manifold with Boundary That Agree at Parameters Point Agree over Common Parameters Area

<The previous article in this series | The table of contents of this series |

description/proof of that for \(C^\infty\) manifold with boundary and \(C^\infty\) vectors field over manifold with boundary, integral curves into interior of manifold with boundary that agree at parameters point agree over common parameters area

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Topics

About: \(C^\infty\) manifold

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The table of contents of this article

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

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Starting Context

• The reader knows a definition of \(C^\infty\) manifold with boundary.
• The reader knows a definition of integral curve of \(C^\infty\) vectors field over \(C^\infty\) manifold with boundary.
• The reader admits 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.
• The reader admits the proposition that the image of any continuous map from any compact topological space to the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum.
• The reader admits the local unique solution existence for a closed interval domain for a Euclidean-normed Euclidean vectors space ordinary differential equation with an initial condition with a clarification on the solution domain area.
• The reader admits the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition.

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Target Context

• The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary and any \(C^\infty\) vectors field over the manifold with boundary, any integral curves into the interior of the manifold with boundary that agree at any parameters point agree over the common parameters area.

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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: Structured Description

Here is the rules of Structured Description.

Entities:
\(M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\((TM, M, \pi)\): \(= \text{ the tangent vectors bundle over } M\)
\(s\): \(\in \{\text{ the } C^\infty \text{ sections of } \pi\}\)
\(J_1\): \(\in \{\text{ the intervals of } \mathbb{R}\}\)
\(J_2\): \(\in \{\text{ the intervals of } \mathbb{R}\}\)
\(J\): \(= J_1 \cap J_2 \subseteq \mathbb{R}\)
\(\gamma_1\): \(: J_1 \to Int (M)\), \(\in \{\text{ the integral curves of } s\}\)
\(\gamma_2\): \(: J_2 \to Int (M)\), \(\in \{\text{ the integral curves of } s\}\)
//

Statements:
\(\exists r_0 \in J (\gamma_1 (r_0) = \gamma_2 (r_0))\)
\(\implies\)
\(\gamma_1 \vert_J = \gamma_2 \vert_J\)
//

\(Int (M)\) means the manifold interior of \(M\).

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2: Note

The reason why this proposition requires \(\gamma_j\) to be into \(Int (M)\) is that otherwise, we would not be able to apply the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition, which requires the domain of \(f\) to be an open subset of \(\mathbb{R}^d\): for any boundary point, \(m \in M\), any chart around \(m\), \((U_m \subseteq M, \phi_m)\), and the induced chart, \((\pi^{-1} (U_m) \subseteq TM, \widetilde{\phi_m})\), \(f = \pi_1 \circ \widetilde{\phi_m} \circ s \circ {\phi_m}^{-1}: \phi_m (U_m) \to \mathbb{R}^d\), where \(\pi_1: \widetilde{\phi_m} (\pi^{-1} (U_m)) = \mathbb{R}^d \times \phi_m (U_m) \to \mathbb{R}^d\) is the projection, would not be any map from open subset of \(\mathbb{R}^d\).

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3: Proof

Whole Strategy: Step 0: suppose that \(J\) is not any 1 point interval hereafter; Step 1: when \(r_0\) is not in the interior of \(J\), expand \(J_1\) or \(J_2\) (or both) to make \(r_0\) to be in the interior of \(J\); Step 2: suppose that \(J\) is not upper bounded; Step 3: suppose that \(\gamma_1 \vert_{[r_0, \infty)} \neq \gamma_2 \vert_{[r_0, \infty)}\), and find a contradiction; Step 4: suppose that \(J\) is upper bounded; Step 5: suppose that \(r' := Sup (\{r \in J \vert r_0 \le r \land \gamma_1 \vert_{[r_0, r]} = \gamma_2 \vert_{[r_0, r]}\})\) was not the upper boundary, and find a contradiction; Step 6: suppose that \(J\) is not lower bounded; Step 7: suppose that \(\gamma_1 \vert_{(- \infty, r_0]} \neq \gamma_2 \vert_{(- \infty, r_0]}\), and find a contradiction; Step 8: suppose that \(J\) is lower bounded; Step 9: suppose that \(r' := Inf (\{r \in J \vert r \le r_0 \land \gamma_1 \vert_{[r, r_0]} = \gamma_2 \vert_{[r, r_0]}\})\) was not the lower boundary, and find a contradiction; Step 10: conclude the proposition.

Step 0:

When \(J\) is any 1 point interval, the proposition obviously holds.

So, let us suppose that \(J\) is not any 1 point interval, hereafter.

Step 1:

When \(r_0\) is not in the interior of \(J\), let us expand \(J_1\) or \(J_2\) (or both) to make \(r_0\) to be in the interior of \(J\).

\(r_0\)'s not in the interior of \(J\) means that \(r_0\) is a closed boundary of \(J_1\) or is a closed boundary of \(J_2\) or both.

Let us suppose that \(r_0\) is a closed boundary of \(J_j\).

Let \(m_0 := \gamma_1 (r_0) = \gamma_2 (r_0)\).

Let us take a interior chart around \(m_0\), \((U_{m_0} \subseteq M, \phi_{m_0})\), and the induced chart, \((\pi^{-1} (U_{m_0}) \subseteq TM, \widetilde{\phi_{m_0}})\).

Let us think of the ordinary differential equation for \(\phi_{m_0} \circ \gamma\), \(\partial_1 (\phi_{m_0} \circ \gamma (r)) = \pi_1 \circ \widetilde{\phi_{m_0}} \circ s \circ {\phi_{m_0}}^{- 1} (\phi_{m_0} \circ \gamma (r))\), where \(\pi_1: \widetilde{\phi_{m_0}} (\pi^{-1} (U_{m_0})) = \mathbb{R}^d \times \phi_{m_0} (U_{m_0}) \to \mathbb{R}^d\) is the projection, with the initial condition, \(\phi_{m_0} \circ \gamma (r_0) = \phi_{m_0} (m_0)\).

\(f: \phi_{m_0} (U_{m_0}) \to \mathbb{R}^d = \pi_1 \circ \widetilde{\phi_{m_0}} \circ s \circ {\phi_{m_0}}^{- 1}\) is \(C^\infty\), because \(s\) is \(C^\infty\).

\(f\) is locally Lipschitz, by 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.

So, there is a \(B_{\phi_{m_0} (m_0), K}\) such that \(\overline{B_{\phi_{m_0} (m_0), K}} \subseteq \phi_{m_0} (U_{m_0})\) and \(f\) is Lipschitz over \(B_{\phi_{m_0} (m_0), K}\) with \(L\).

\(f \le M'\) for an \(M' \in \mathbb{R}\) over \(\overline{B_{\phi_{m_0} (m_0), K}}\), so, over \(B_{\phi_{m_0} (m_0), K}\), by the proposition that 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, by the local unique solution existence for a closed interval domain for a Euclidean-normed Euclidean vectors space ordinary differential equation with an initial condition with a clarification on the solution domain area, there is the unique solution for the equation over \([r_0 - \epsilon_1, r_0 + \epsilon_2]\), where \(0 \lt \epsilon_1, \epsilon_2\), \(\phi_{m_0} \circ \gamma: [r_0 - \epsilon_1, r_0 + \epsilon_2] \to B_{\phi_{m_0} (m_0), K}\).

That means that \(d \gamma / d r (r) = s (\gamma (r))\), so, \(\gamma\) is an integral curve of \(s\).

\(\phi_{m_0} \circ \gamma \vert_{J_j \cap [r_0 - \epsilon_1, r_0 + \epsilon_2]} = \phi_{m_0} \circ \gamma_j \vert_{J_j \cap [r_0 - \epsilon_1, r_0 + \epsilon_2]}\), because \(\phi_{m_0} \circ \gamma \vert_{J_j \cap [r_0 - \epsilon_1, r_0 + \epsilon_2]}\) and \(\phi_{m_0} \circ \gamma_j \vert_{J_j \cap [r_0 - \epsilon_1, r_0 + \epsilon_2]}\) are some solutions of the ordinary differential equation with initial condition while the solution is unique.

That means that \(\gamma \vert_{J_j \cap [r_0 - \epsilon_1, r_0 + \epsilon_2]} = \gamma_j \vert_{J_j \cap [r_0 - \epsilon_1, r_0 + \epsilon_2]}\).

So, \(\gamma_j\) can be extended over \(J_j \cup [r_0 - \epsilon_1, r_0 + \epsilon_2]\) such that it is the original \(\gamma_j\) over \(J_j\) and it is \(\gamma\) over \([r_0 - \epsilon_1, r_0 + \epsilon_2]\).

Let \(J_j \cup [r_0 - \epsilon_1, r_0 + \epsilon_2]\) be denoted as \(J_j\) hereafter.

Now, \(r_0\) is in the interior of \(J_j\).

If the proposition is proved for the wider \(J_j\) s, the proposition is true for the original \(J_j\) s, because the \(J\) for the latter is contained in the \(J\) for the former.

So, hereafter, we suppose that \(r_0\) is in the interior of \(J\).

Step 2:

\(J\) may be not upper bounded or be upper bounded.

Let us suppose that \(J\) is not upper bounded.

Step 3:

Let us suppose that \(\gamma_1 \vert_{[r_0, \infty)} \neq \gamma_2 \vert_{[r_0, \infty)}\).

Let us define \(r' := Sup (\{r \in J \vert r_0 \le r \land \gamma_1 \vert_{[r_0, r]} = \gamma_2 \vert_{[r_0, r]}\})\), where the supremum was taken in \(\mathbb{R}\), which would exist as \(r' \in \mathbb{R}\), by the supposition.

\(r' \in Int (J)\), obviously.

\(\gamma_1 (r') = \gamma_2 (r')\), because if \(r' = r_0\), \(\gamma_1 (r') = \gamma_1 (r_0) = \gamma_2 (r_0) = \gamma_2 (r')\), and otherwise, if \(\gamma_1 (r') \neq \gamma_2 (r')\), there would be an open neighborhood of \(\gamma_1 (r')\), \(U_{\gamma_1 (r')} \subseteq M\), and an open neighborhood of \(\gamma_2 (r')\), \(U_{\gamma_2 (r')} \subseteq M\), such that \(U_{\gamma_1 (r')} \cap U_{\gamma_2 (r')} = \emptyset\), because \(M\) was Hausdorff, and there would be a \(B_{r', \delta} \subseteq (r_0, \infty)\) such that \(\gamma_1 (B_{r', \delta}) \subseteq U_{\gamma_1 (r')}\) and \(\gamma_2 (B_{r', \delta}) \subseteq U_{\gamma_2 (r')}\), because \(\gamma_j\) was continuous, which would mean that \(\gamma_1 (B_{r', \delta}) \cap \gamma_2 (B_{r', \delta}) = \emptyset\), which would mean that \(\gamma_1 (r' - \delta / 2) \neq \gamma_2 (r' - \delta / 2)\), a contradiction against that \(r'\) was the supremum.

So, let \(m' := \gamma_1 (r') = \gamma_2 (r')\).

There would be a interior chart around \(m'\), \((U_{m'} \subseteq M, \phi_{m'})\), and the induced chart, \((\pi^{-1} (U_{m'}) \subseteq TM, \widetilde{\phi_{m'}})\).

As \(\gamma_j \vert_J\) was continuous and \(r' \in Int (J)\), there would be a positive \(\delta\) such that \(B_{r', \delta} \subseteq J\) and \(\gamma_j (B_{r', \delta}) \subseteq U_{m'}\). Then, \(\phi_{m'} \circ \gamma_j (B_{r', \delta}) \subseteq \phi_{m'} (U_{m'})\).

As \(\gamma_j\) was an integral curve of \(s\), over \(B_{r', \delta}\), \(d \gamma_j / d r (r) = s (\gamma_j (r))\), which is components-wise, \(\partial_1 (\phi_{m'} \circ \gamma_j (r)) = \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} \circ \phi_{m'} \circ \gamma_j (r)\), where \(\pi_1: \widetilde{\phi_{m'}} (\pi^{-1} (U_{m'})) = \mathbb{R}^d \times \phi_{m'} (U_{m'}) \to \mathbb{R}^d\) is the projection, with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which is the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(f: \phi_{m'} (U_{m'}) \to \mathbb{R}^d, p \mapsto \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} (p)\), \(C^\infty\), because \(s\) is \(C^\infty\).

\(f\) is locally Lipschitz, by 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.

So, by the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition, \(\gamma_1 \vert_{B_{r', \delta}} = \gamma_2 \vert_{B_{r', \delta}}\), which would mean that \(\gamma_1 \vert_{[r_0, r' + \delta / 2]} = \gamma_2 \vert_{[r_0, r' + \delta / 2]}\), a contradiction against that \(r'\) was the supremum.

So, \(\gamma_1 \vert_{[r_0, \infty)} = \gamma_2 \vert_{[r_0, \infty)}\).

Step 4:

Let us suppose that \(J\) is upper bounded with the upper boundary, \(r_2\).

Step 5:

Note that \(J\) may be upper open or upper closed.

What we will do is parallel to the case that \(J\) is not upper bounded.

Let us define \(r' := Sup (\{r \in J \vert r_0 \le r \land \gamma_1 \vert_{[r_0, r]} = \gamma_2 \vert_{[r_0, r]}\})\), where the supremum is taken in \(\mathbb{R}\), which exists as \(r' \in \mathbb{R}\), because \(J\) is upper bounded.

Let us suppose that \(r'\) was not the upper boundary, which would mean \(r' \lt r_2\).

\(r' \in Int (J)\) whether \(J\) is upper open or upper closed, obviously.

\(\gamma_1 (r') = \gamma_2 (r')\), because if \(r' = r_0\), \(\gamma_1 (r') = \gamma_1 (r_0) = \gamma_2 (r_0) = \gamma_2 (r')\), and otherwise, if \(\gamma_1 (r') \neq \gamma_2 (r')\), there would be an open neighborhood of \(\gamma_1 (r')\), \(U_{\gamma_1 (r')} \subseteq M\), and an open neighborhood of \(\gamma_2 (r')\), \(U_{\gamma_2 (r')} \subseteq M\), such that \(U_{\gamma_1 (r')} \cap U_{\gamma_2 (r')} = \emptyset\), because \(M\) was Hausdorff, and there would be a \(B_{r', \delta} \subseteq (r_0, r_2)\) such that \(\gamma_1 (B_{r', \delta}) \subseteq U_{\gamma_1 (r')}\) and \(\gamma_2 (B_{r', \delta}) \subseteq U_{\gamma_2 (r')}\), because \(\gamma_j\) was continuous, which would mean that \(\gamma_1 (B_{r', \delta}) \cap \gamma_2 (B_{r', \delta}) = \emptyset\), which would mean that \(\gamma_1 (r' - \delta / 2) \neq \gamma_2 (r' - \delta / 2)\), a contradiction against that \(r'\) was the supremum.

So, let \(m' := \gamma_1 (r') = \gamma_2 (r')\).

There would be a interior chart around \(m'\), \((U_{m'} \subseteq M, \phi_{m'})\), and the induced chart, \((\pi^{-1} (U_{m'}) \subseteq TM, \widetilde{\phi_{m'}})\).

As \(\gamma_j \vert_J\) was continuous and \(r' \in Int (J)\), there would be a positive \(\delta\) such that \(B_{r', \delta} \subseteq J\) and \(\gamma_j (B_{r', \delta}) \subseteq U_{m'}\). Then, \(\phi_{m'} \circ \gamma_j (B_{r', \delta}) \subseteq \phi_{m'} (U_{m'})\).

As \(\gamma_j\) was an integral curve of \(s\), over \(B_{r', \delta}\), \(d \gamma_j / d r (r) = s (\gamma_j (r))\), which is components-wise, \(\partial_1 (\phi_{m'} \circ \gamma_j (r)) = \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} \circ \phi_{m'} \circ \gamma_j (r)\), where \(\pi_1: \widetilde{\phi_{m'}} (\pi^{-1} (U_{m'})) = \mathbb{R}^d \times \phi_{m'} (U_{m'}) \to \mathbb{R}^d\) is the projection, with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which is the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(f: \phi_{m'} (U_{m'}) \to \mathbb{R}^d, p \mapsto \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} (p)\), \(C^\infty\), because \(s\) is \(C^\infty\).

\(f\) is locally Lipschitz, by 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.

So, by the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition, \(\gamma_1 \vert_{B_{r', \delta}} = \gamma_2 \vert_{B_{r', \delta}}\), which would mean that \(\gamma_1 \vert_{[r_0, r' + \delta / 2]} = \gamma_2 \vert_{[r_0, r' + \delta / 2]}\), a contradiction against that \(r'\) was the supremum.

So, \(r'\) is the upper boundary, which means \(r' = r_2\).

That means that \(\gamma_1 \vert_{[r_0, r_2)} = \gamma_2 \vert_{[r_0, r_2)}\) or \(\gamma_1 \vert_{[r_0, r_2]} = \gamma_2 \vert_{[r_0, r_2]}\) according to whether \(J\) is upper open or is upper closed: \(\gamma_1 (r_2) = \gamma_2 (r_2)\) can be proved as before.

Step 6:

For the area smaller than \(r_0\), the logic is parallel.

\(J\) may be not lower bounded or be lower bounded.

Let us suppose that \(J\) is not lower bounded.

Step 7:

Let us suppose that \(\gamma_1 \vert_{(- \infty, r_0]} \neq \gamma_2 \vert_{(- \infty, r_0]}\).

Let us define \(r' := Inf (\{r \in J \vert r \le r_0 \land \gamma_1 \vert_{[r, r_0]} = \gamma_2 \vert_{[r, r_0]}\})\), where the infimum was taken in \(\mathbb{R}\), which would exist as \(r' \in \mathbb{R}\), by the supposition.

\(r' \in Int (J)\), obviously.

\(\gamma_1 (r') = \gamma_2 (r')\), because if \(r' = r_0\), \(\gamma_1 (r') = \gamma_1 (r_0) = \gamma_2 (r_0) = \gamma_2 (r')\), and otherwise, if \(\gamma_1 (r') \neq \gamma_2 (r')\), there would be an open neighborhood of \(\gamma_1 (r')\), \(U_{\gamma_1 (r')} \subseteq M\), and an open neighborhood of \(\gamma_2 (r')\), \(U_{\gamma_2 (r')} \subseteq M\), such that \(U_{\gamma_1 (r')} \cap U_{\gamma_2 (r')} = \emptyset\), because \(M\) was Hausdorff, and there would be a \(B_{r', \delta} \subseteq (- \infty, r_0)\) such that \(\gamma_1 (B_{r', \delta}) \subseteq U_{\gamma_1 (r')}\) and \(\gamma_2 (B_{r', \delta}) \subseteq U_{\gamma_2 (r')}\), because \(\gamma_j\) was continuous, which would mean that \(\gamma_1 (B_{r', \delta}) \cap \gamma_2 (B_{r', \delta}) = \emptyset\), which would mean that \(\gamma_1 (r' + \delta / 2) \neq \gamma_2 (r' + \delta / 2)\), a contradiction against that \(r'\) was the infimum.

So, let \(m' := \gamma_1 (r') = \gamma_2 (r')\).

There would be a interior chart around \(m'\), \((U_{m'} \subseteq M, \phi_{m'})\), and the induced chart, \((\pi^{-1} (U_{m'}) \subseteq TM, \widetilde{\phi_{m'}})\).

As \(\gamma_j \vert_J\) was continuous and \(r' \in Int (J)\), there would be a positive \(\delta\) such that \(B_{r', \delta} \subseteq J\) and \(\gamma_j (B_{r', \delta}) \subseteq U_{m'}\). Then, \(\phi_{m'} \circ \gamma_j (B_{r', \delta}) \subseteq \phi_{m'} (U_{m'})\).

As \(\gamma_j\) was an integral curve of \(s\), over \(B_{r', \delta}\), \(d \gamma_j / d r (r) = s (\gamma_j (r))\), which is components-wise, \(\partial_1 (\phi_{m'} \circ \gamma_j (r)) = \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} \circ \phi_{m'} \circ \gamma_j (r)\) with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which is the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(f: \phi_{m'} (U_{m'}) \to \mathbb{R}^d, p \mapsto \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} (p)\), \(C^\infty\), because \(s\) is \(C^\infty\).

\(f\) is locally Lipschitz, by 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.

So, by the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition, \(\gamma_1 \vert_{B_{r', \delta}} = \gamma_2 \vert_{B_{r', \delta}}\), which would mean that \(\gamma_1 \vert_{[r' - \delta / 2, r_0]} = \gamma_2 \vert_{[r' - \delta / 2, r_0]}\), a contradiction against that \(r'\) was the infimum.

So, \(\gamma_1 \vert_{(- \infty, r_0]} = \gamma_2 \vert_{(- \infty, r_0]}\).

Step 8:

Let us suppose that \(J\) is lower bounded with the lower boundary, \(r_1\).

Step 9:

Note that \(J\) may be lower open or lower closed.

What we will do is parallel to the case that \(J\) is not lower bounded.

Let us define \(r' := Inf (\{r \in J \vert r \le r_0 \land \gamma_1 \vert_{[r, r_0]} = \gamma_2 \vert_{[r, r_0]}\})\), where the infimum is taken in \(\mathbb{R}\), which exists as \(r' \in \mathbb{R}\), because \(J\) is lower bounded.

Let us suppose that \(r'\) was not the lower boundary, which would mean \(r_1 \lt r'\).

\(r' \in Int (J)\) whether \(J\) is lower open or lower closed, obviously.

\(\gamma_1 (r') = \gamma_2 (r')\), because if \(r' = r_0\), \(\gamma_1 (r') = \gamma_1 (r_0) = \gamma_2 (r_0) = \gamma_2 (r')\), and otherwise, if \(\gamma_1 (r') \neq \gamma_2 (r')\), there would be an open neighborhood of \(\gamma_1 (r')\), \(U_{\gamma_1 (r')} \subseteq M\), and an open neighborhood of \(\gamma_2 (r')\), \(U_{\gamma_2 (r')} \subseteq M\), such that \(U_{\gamma_1 (r')} \cap U_{\gamma_2 (r')} = \emptyset\), because \(M\) was Hausdorff, and there would be a \(B_{r', \delta} \subseteq (r_1, r_0)\) such that \(\gamma_1 (B_{r', \delta}) \subseteq U_{\gamma_1 (r')}\) and \(\gamma_2 (B_{r', \delta}) \subseteq U_{\gamma_2 (r')}\), because \(\gamma_j\) was continuous, which would mean that \(\gamma_1 (B_{r', \delta}) \cap \gamma_2 (B_{r', \delta}) = \emptyset\), which would mean that \(\gamma_1 (r' + \delta / 2) \neq \gamma_2 (r' + \delta / 2)\), a contradiction against that \(r'\) was the infimum.

So, let \(m' := \gamma_1 (r') = \gamma_2 (r')\).

There would be a interior chart around \(m'\), \((U_{m'} \subseteq M, \phi_{m'})\), and the induced chart, \((\pi^{-1} (U_{m'}) \subseteq TM, \widetilde{\phi_{m'}})\).

As \(\gamma_j \vert_J\) was continuous and \(r' \in Int (J)\), there would be a positive \(\delta\) such that \(B_{r', \delta} \subseteq J\) and \(\gamma_j (B_{r', \delta}) \subseteq U_{m'}\). Then, \(\phi_{m'} \circ \gamma_j (B_{r', \delta}) \subseteq \phi_{m'} (U_{m'})\).

As \(\gamma_j\) was an integral curve of \(s\), over \(B_{r', \delta}\), \(d \gamma_j / d r (r) = s (\gamma_j (r))\), which is components-wise, \(\partial_1 (\phi_{m'} \circ \gamma_j (r)) = \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} \circ \phi_{m'} \circ \gamma_j (r)\), where \(\pi_1: \widetilde{\phi_{m'}} (\pi^{-1} (U_{m'})) = \mathbb{R}^d \times \phi_{m'} (U_{m'}) \to \mathbb{R}^d\) is the projection, with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which is the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(f: \phi_{m'} (U_{m'}) \to \mathbb{R}^d, p \mapsto \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} (p)\), \(C^\infty\), because \(s\) is \(C^\infty\).

\(f\) is locally Lipschitz, by 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.

So, by the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition, \(\gamma_1 \vert_{B_{r', \delta}} = \gamma_2 \vert_{B_{r', \delta}}\), which would mean that \(\gamma_1 \vert_{[r' - \delta / 2, r_0]} = \gamma_2 \vert_{[r' - \delta / 2, r_0]}\), a contradiction against that \(r'\) was the infimum.

So, \(r'\) is the lower boundary, which means \(r' = r_1\).

That means that \(\gamma_1 \vert_{(r_1, r_0]} = \gamma_2 \vert_{(r_1, r_0]}\) or \(\gamma_1 \vert_{[r_1, r_0]} = \gamma_2 \vert_{[r_1, r_0]}\) according to whether \(J\) is lower open or is lower closed: \(\gamma_1 (r_1) = \gamma_2 (r_1)\) can be proved as before.

Step 10:

So, \(\gamma_1 \vert_J = \gamma_2 \vert_J\).

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References

<The previous article in this series | The table of contents of this series |

1641: \(C^k\)-ness of Solution for Euclidean-Normed Euclidean Vectors Space ODE with Initial Condition

<The previous article in this series | The table of contents of this series | The next article in this series>

description/proof of the \(C^k\)-ness of solution for Euclidean-normed Euclidean vectors space ODE with initial condition

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Topics

About: vectors space

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The table of contents of this article

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

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Starting Context

• The reader knows a definition of Euclidean-normed Euclidean vectors space.
• The reader knows a definition of derivative of map from arbitrary subset of normed vectors space into subset of normed vectors space at point.
• The reader knows a definition of continuous, topological spaces map.
• The reader knows a definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
• The reader knows a definition of matrix norm induced by vector norms.
• The reader knows a definition of Frobenius matrix norm.
• The reader knows a definition of uniformly Cauchy sequence of maps from set into metric space.
• 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 for any topological space, any compact subset of the space that is contained in any subspace is compact on the subspace.
• The reader admits the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.
• The reader admits the proposition that the product of any finite number of compact topological spaces is compact.
• The reader admits the proposition that for any possibly uncountable number of indexed topological spaces or any finite number of topological spaces and their subspaces, the product of the subspaces is the subspace of the product of the base spaces.
• The reader admits the proposition that for any continuous map between any topological spaces, the image of any compact subset of the domain is a compact subset of the codomain.
• The reader admits the proposition that for any metric space, any open subset, and any compact subset contained in the open subset, there is a positive radius of which the open or closed ball around each point on the compact subset is contained in the open subset.
• The reader admits the proposition that for any topological space, any compact subset of any subspace is compact on the base space.
• The reader admits the proposition that for any metric space, any bounded subset, and any positive real number, the union of the open balls around each point of the subset with the number radius is bounded.
• The reader admits the proposition that for any metric space with the induced topology, the closure of any bounded subset is bounded with the diameter of the subset.
• The reader admits the proposition that for any metric space, any subset, any subset of the subset, and any positive real number, if the open ball around each point of the subset of the subset with the number-radius is contained in the subset, the closure of the union of the open balls around each point of the subset of the subset with any smaller-radius is contained in the subset.
• The reader admits the proposition that for any continuous map with any compact topological parameter space from any subspace of any Euclidean metric space with the induced topology into any any subspace of any Euclidean metric space with the induced topology, that (the map) locally satisfies Lipschitz estimates, the restriction of the map on any compact subspace domain satisfies Lipschitz estimates.
• The reader admits the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
• The reader admits the proposition that the image of any continuous map from any compact topological space to the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum.
• The reader admits the proposition that for any continuous real map from any non-1-point interval with any closed end, if the image of the interior is bounded, the range is correspondingly equal-or-smaller-or-larger bounded.
• The reader admits the local unique solution existence for a closed interval domain for a Euclidean-normed Euclidean vectors space ordinary differential equation with an initial condition with a clarification on the solution domain area.
• The reader admits the proposition that for any uniformly convergent sequence of continuous maps from any topological space into any metric space with the induced topology, the convergence is continuous.
• The reader admits the mean-value theorem for any differentiable map from any closed interval into the \(1\)-dimensional Euclidean \(C^\infty\) manifold.
• The reader admits the proposition that for any continuous map between any metric spaces with the domain with the induced topology, the restriction of the map on any compact domain is uniformly continuous.
• The reader admits the proposition that for any differentiable map from any open interval into any Euclidean \(C^\infty\) manifold with the Euclidean norm, the norm of each point image is upper-bounded by a differentiable map between the non-negative interval with an initial condition if there is a Lipschitz map between the non-negative interval that satisfies certain conditions.
• The reader admits the proposition that any uniformly Cauchy sequence of maps from any set into any complete metric space converges uniformly.
• The reader admits the proposition that for any maps between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.

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Target Context

• The reader will have a description and a proof of the \(C^k\)-ness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition.

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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: Structured Description

Here is the rules of Structured Description.

Entities:
\(\mathbb{R}^d\): \(= \text{ the Euclidean-normed Euclidean vectors space }\) as the Euclidean \(C^\infty\) manifold
\(\mathbb{R}\): \(= \text{ the Euclidean-normed Euclidean vectors space }\) as the Euclidean \(C^\infty\) manifold
\(U\): \(\in \{\text{ the open subsets of } \mathbb{R}^d\}\)
\(J\): \(\in \{\text{ the intervals of } \mathbb{R}\}\) as the Euclidean \(C^\infty\) manifold with boundary
\(f\): \(: U \times J \to \mathbb{R}^d\), \(\in \{\text{ the } C^k \text{ maps }\}\), such that \(\forall u \in U (\exists U_u \in \{\text{ the open neighborhoods of } u \text{ on } U\} (\forall u_1, u_2 \in U_u, \forall r \in J (\Vert f (u_1, r) - f (u_2, r) \Vert \le L_u \Vert u_1 - u_2 \Vert)))\)
\(r_0\): \(\in J\)
//

Statements:
\(\exists J^` \in \{\text{ the open intervals of } J\} \text{ such that } r_0 \in J^`, \exists U^` \in \{\text{ the open subsets of } U\}, \exists x: J^` \times U^` \to U (\partial_1 x (r, p) = f (x (r, p), r) \land x (r_0, p) = p)\)
\(\implies\)
\(x \in \{\text{ the } C^k \text{ maps }\}\)
//

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2: Proof

Whole Strategy: prove it inductively with respect to \(k\); Step 1: take any \((r_1, p_1) \in J^` \times U^`\), any bounded open interval around \(r_1\), \(J_{r_1}\), such that \(r_0, r_1 \in J_{r_1}\) and \(\overline{J_{r_1}} \subseteq J^`\), \(K := x (\overline{J_{r_1}} \times \{p_1\})\), and \(V := \cup_{k \in K} B_{k, \epsilon} \subseteq U\); Step 2: take Lipschitz \(f \vert_{\overline{V} \times \overline{J_{r_1}}}\) with \(L\), define \(M := Sup (\{\Vert f (p, r) \Vert \vert (p, r) \in \overline{V} \times \overline{J_{r_1}}\})\), \(R := Sup (\{\Vert r - r_0 \Vert \vert r \in \overline{J_{r_1}}\})\), and \(\delta\) such that \(2 \delta e^{L R} \lt \epsilon\) and \(B_{p_1, 2 \delta} \subseteq U^`\), and see that \(x\) maps \(\overline{J_{r_1}} \times \overline{B_{p_1, 2 \delta}}\) into \(V\); Step 3: prove it for \(k = 0\); Step 4: prove it for \(k = 1\); Step 5: suppose that it holds for \(0 \le k \le k' - 1\) where \(2 \le k'\), and prove it for \(k = k'\); Step 6: conclude the proposition.

Step 1:

Let \((r_1, p_1) \in J^` \times U^`\) be any.

Let \(J_{r_1}\) be any bounded open interval around \(r_1\) such that \(r_0, r_1 \in J_{r_1}\) and \(\overline{J_{r_1}} \subseteq J^`\), which is possible, because \(J^`\) is an open interval.

\(x \vert_{J^` \times \{p_1\}}: J^` \times \{p_1\} \to U\) is continuous, because \(x\) is differentiable with respect to the 1st argument.

\(\overline{J_{r_1}} \subseteq J^`\) is a compact subset, by the Heine-Borel theorem: any subset of any Euclidean topological space is compact if and only if it is closed and bounded and the proposition that for any topological space, any compact subset of the space that is contained in any subspace is compact on the subspace: \(\overline{J_{r_1}}\) is compact on \(\mathbb{R}\) and is compact on \(J^`\).

\(\overline{J_{r_1}} \subseteq J^`\) is a compact subspace, by the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.

\(\{p_1\}\) is a compact topological space, because for any open cover of \(\{p_1\}\), there is the 1-element subcover that take any element that contains \(p_1\).

\(\overline{J_{r_1}} \times \{p_1\}\) is a compact topological space, by the proposition that the product of any finite number of compact topological spaces is compact.

\(\overline{J_{r_1}} \times \{p_1\}\) is the topological subspace of \(J^` \times \{p_1\}\), by the proposition that for any possibly uncountable number of indexed topological spaces or any finite number of topological spaces and their subspaces, the product of the subspaces is the subspace of the product of the base spaces.

\(\overline{J_{r_1}} \times \{p_1\}\) is a compact subset of \(J^` \times \{p_1\}\), by the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.

\(K := x (\overline{J_{r_1}} \times \{p_1\})\) is compact on \(U\), by the proposition that for any continuous map between any topological spaces, the image of any compact subset of the domain is a compact subset of the codomain.

There is an \(\epsilon \in \mathbb{R}\) such that \(0 \lt \epsilon\) and for each \(k \in K\), \(B'_{k, 2 \epsilon} \subseteq U\), by the proposition that for any metric space, any open subset, and any compact subset contained in the open subset, there is a positive radius of which the open or closed ball around each point on the compact subset is contained in the open subset.

Let \(V := \cup_{k \in K} B_{k, \epsilon} \subseteq U\).

\(V\) is an open neighborhood of \(K\) on \(U\).

\(K\) is a compact subset of \(\mathbb{R}^d\), by the proposition that for any topological space, any compact subset of any subspace is compact on the base space, so, is bounded, by the Heine-Borel theorem: any subset of any Euclidean topological space is compact if and only if it is closed and bounded.

\(V\) is bounded on \(\mathbb{R}^d\), by the proposition that for any metric space, any bounded subset, and any positive real number, the union of the open balls around each point of the subset with the number radius is bounded.

\(\overline{V}\) on \(\mathbb{R}^d\) is bounded, by the proposition that for any metric space with the induced topology, the closure of any bounded subset is bounded with the diameter of the subset.

So, \(\overline{V}\) is compact on \(\mathbb{R}^d\), by the Heine-Borel theorem: any subset of any Euclidean topological space is compact if and only if it is closed and bounded.

\(\overline{V} \subseteq U\), by the proposition that for any metric space, any subset, any subset of the subset, and any positive real number, if the open ball around each point of the subset of the subset with the number-radius is contained in the subset, the closure of the union of the open balls around each point of the subset of the subset with any smaller-radius is contained in the subset.

\(\overline{V}\) is compact on \(U\) as the topological subspace, by the proposition that for any topological space, any compact subset of the space that is contained in any subspace is compact on the subspace.

Step 2:

\(f \vert_{\overline{V} \times \overline{J_{r_1}}}\) is a Lipschitz map with \(L\), which depends on \(V\), but we denote it just as \(L\), by the proposition that for any continuous map with any compact topological parameter space from any subspace of any Euclidean metric space with the induced topology into any any subspace of any Euclidean metric space with the induced topology, that (the map) locally satisfies Lipschitz estimates, the restriction of the map on any compact subspace domain satisfies Lipschitz estimates.

\(\overline{V} \times \overline{J_{r_1}}\) is a compact topological space, by the proposition that the product of any finite number of compact topological spaces is compact.

\(\overline{V} \times \overline{J_{r_1}}\) is the topological subspace of \(U \times J\), by the proposition that for any possibly uncountable number of indexed topological spaces or any finite number of topological spaces and their subspaces, the product of the subspaces is the subspace of the product of the base spaces, so, it is a compact subspace.

\(f \vert_{\overline{V} \times \overline{J_{r_1}}}: \overline{V} \times \overline{J_{r_1}} \to \mathbb{R}^d\) is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.

Let \(M := Sup (\{\Vert f (p, r) \Vert \vert (p, r) \in \overline{V} \times \overline{J_{r_1}}\}) \lt \infty\), by the proposition that the image of any continuous map from any compact topological space to the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum.

Let \(R := Sup (\{\Vert r - r_0 \Vert \vert r \in \overline{J_{r_1}}\}) \lt \infty\), because \(J_{r_1}\) is bounded.

Let \(\delta \in \mathbb{R}\) be any such that \(0 \lt \delta\), \(2 \delta e^{L R} \lt \epsilon\) (which implies \(2 \delta \lt \epsilon\)), and \(\overline{B_{p_1, 2 \delta}} \subseteq U^`\), which is possible, because \(U^`\) is an open neighborhood of \(p_1\): there is a \(B_{p_1, \delta'} \subseteq U^`\), so, take \(2 \delta \lt \delta'\), then, \(\overline{B_{p_1, 2 \delta}} \subseteq B_{p_1, \delta'} \subseteq U^`\).

For each \(p, p' \in U^`\), if there is an open interval around \(r_0\), \(J_{r_0} \subset \overline{J_{r_1}}\), such that \(x (r, p), x (r, p') \in \overline{V}\) over it, let us see that we can take \(u: J_{r_0} \to \mathbb{R}^d, r \mapsto x (r, p) - x (r, p')\), \(v: [0, \infty) \to [0, \infty), r \mapsto e^{L r} \Vert x (r_0, p) - x (r_0, p') \Vert\), and \(w: [0, \infty) \to [0, \infty), r \mapsto L r\), which satisfy the conditions for the proposition that for any differentiable map from any open interval into any Euclidean \(C^\infty\) manifold with the Euclidean norm, the norm of each point image is upper-bounded by a differentiable map between the non-negative interval with an initial condition if there is a Lipschitz map between the non-negative interval that satisfies certain conditions for \(u, v, w\) instead of \(f, g, h\).

\(u\) is differentiable; \(v\) is differentiable and satisfies \(v (0) = \Vert x (r_0, p) - x (r_0, p') \Vert = \Vert u (r_0) \Vert\); \(w\) is a Lipschitz map with \(L\) and satisfies \(\Vert \partial_1 u (r) \Vert = \Vert f (x (r, p), r) - f (x (r, p'), r) \Vert \le L \Vert x (r, p) - x (r, p') \Vert = w (\Vert u (r) \Vert)\) and \(\partial_1 v (r) = L e^{L r} \Vert x (r_0, p) - x (r_0, p') \Vert = w (v (r))\).

So, in that case, \(\Vert u (r) \Vert = \Vert x (r, p) - x (r, p') \Vert \le v (\vert r - r_0 \vert) = e^{L \vert r - r_0 \vert} \Vert x (r_0, p) - x (r_0, p') \Vert \le e^{L R} \Vert p - p' \Vert\).

Let us see that \(x (\overline{J_{r_1}} \times \overline{B_{p_1, 2 \delta}}) \subseteq V\).

Let us suppose that there was a \((r_2, p_2) \in \overline{J_{r_1}} \times \overline{B_{p_1, 2 \delta}}\) such that \(x (r_2, p_2) \notin V\).

\(p_2 \in V\), because \(x (r_0, p_2) = p_2 \in \overline{B_{p_1, 2 \delta}} \subseteq B_{p_1, \epsilon}\) and \(p_1 = x (r_0, p_1) \in K\).

\(r_2 \neq r_0\), because \(x (r_0, p_2) = p_2 \in V\).

Let us suppose that \(r_0 \lt r_2\).

Let \(t := Inf (\{r \in \overline{J_{r_1}} \vert r_0 \lt r \land x (r, p_2) \notin V\})\), which would exist as \(r_0 \le t\), because at least \(x (r_2, p_2) \notin V\).

\(x (t, p_2) \notin V\), because if \(t\) was the upper boundary of \(\overline{J_{r_1}}\), \(t = r_2\), which would mean that \(x (t, p_2) = x (r_2, p_2) \notin V\), and otherwise, if \(x (t, p_2) \in V\), as \(V\) was open and \(x\) was continuous with respect to \(r\), there would be a \(B_{t, \rho} \subseteq J_{r_1}\) such that \(x (B_{t, \rho}, p_2) \subseteq V\), then, \(t\) would not be the infimum, a contradiction.

So, \(t \neq r_0\), so, \(r_0 \lt t\).

For each \(r \in [r_0, t)\), \(x (r, p_2) \in V\), because otherwise, \(t\) would not be the infimum, a contradiction; also \(x (r, p_1) \in V\), because \(x (r, p_1) \in K \subseteq V\).

So, there would be an open interval, \(J_{r_0}\), that contained \([r_0, t)\) such that for each \(r \in J_{r_0}\), \(x (r, p_1), x (r, p_2) \in V\).

So, for each \(r \in J_{r_0}\), \(\Vert x (r, p_2) - x (r, p_1) \Vert \le e^{L R} \Vert p_2 - p_1 \Vert \le e^{L R} 2 \delta \lt \epsilon\).

As \(x\) was continuous with respect to \(r\), \(\Vert x (t, p_2) - x (t, p_1) \Vert \le e^{L R} 2 \delta \lt \epsilon\), by the proposition that for any continuous real map from any non-1-point interval with any closed end, if the image of the interior is bounded, the range is correspondingly equal-or-smaller-or-larger bounded, which would mean that \(x (t, p_2) \in V\), because \(x (t, p_1) \in K\), a contradiction.

So, there is no such \((r_2, p_2)\) such that \(r_0 \le r_2\).

Let us suppose that \(r_2 \lt r_0\).

Let \(t := Sup (\{r \in \overline{J_{r_1}} \vert r \lt r_0 \land x (r, p_2) \notin V\})\), which would exist as \(t \le r_0\), because at least \(x (r_2, p_2) \notin V\).

\(x (t, p_2) \notin V\), because if \(t\) was the lower boundary of \(\overline{J_{r_1}}\), \(t = r_2\), which would mean that \(x (t, p_2) = x (r_2, p_2) \notin V\), and otherwise, if \(x (t, p_2) \in V\), as \(V\) was open and \(x\) was continuous with respect to \(r\), there would be a \(B_{t, \rho} \subseteq J_{r_1}\) such that \(x (B_{t, \rho}, p_2) \subseteq V\), then, \(t\) would not be the supremum, a contradiction.

So, \(t \neq r_0\), so, \(t \lt r_0\).

For each \(r \in (t, r_0]\), \(x (r, p_2) \in V\), because otherwise, \(t\) would not be the supremum, a contradiction; also \(x (r, p_1) \in V\), because \(x (r, p_1) \in K \subseteq V\).

So, there would be a \(J_{r_0}\) that contained \((t, r_0]\) such that for each \(r \in J_{r_0}\), \(x (r, p_1), x (r, p_2) \in V\).

So, for each \(r \in J_{r_0}\), \(\Vert x (r, p_2) - x (r, p_1) \Vert \le e^{L R} \Vert p_2 - p_1 \Vert \le e^{L R} 2 \delta \lt \epsilon\).

As \(x\) was continuous with respect to \(r\), \(\Vert x (t, p_2) - x (t, p_1) \Vert \le e^{L R} 2 \delta \lt \epsilon\), by the proposition that for any continuous real map from any non-1-point interval with any closed end, if the image of the interior is bounded, the range is correspondingly equal-or-smaller-or-larger bounded, which would mean that \(x (t, p_2) \in V\), because \(x (t, p_1) \in K\), a contradiction.

So, there is no such \((r_2, p_2)\) such that \(r_2 \lt r_0\).

So, there is no such whatsoever \((r_2, p_2)\).

So, \(x (\overline{J_{r_1}} \times \overline{B_{p_1, 2 \delta}}) \subseteq V\).

Step 3:

Let us prove the proposition for \(k = 0\).

By Step 2, \(x\) from \(\overline{J_{r_1}} \times \overline{B_{p_1, 2 \delta}}\) is into \(V\), so, for each \(r \in J_{r_1}\) (\(J_{r_1}\) is taken as \(J_{r_0}\)) and \(p, p' \in \overline{B_{p_1, 2 \delta}}\), \(\Vert x (r, p) - x (r, p') \Vert \le e^{L R} \Vert p - p' \Vert\).

For each \(r \in J_{r_1}\) and \(p \in \overline{B_{p_1, 2 \delta}} \subseteq U^`\), \(x (r, p) = p + \int^r_{r_0} f (x (s, p), s) d s\), which has been proved in Step 1 of Proof of the local unique solution existence for a closed interval domain for a Euclidean-normed Euclidean vectors space ordinary differential equation with an initial condition with a clarification on the solution domain area.

\(\Vert x (r, p) - x (r_1, p_1) \Vert = \Vert p - p_1 + \int^r_{r_0} f (x (s, p), s) d s - \int^{r_1}_{r_0} f (x (s, p_1), s) d s \Vert = \Vert p - p_1 + \int^{r_1}_{r_0} (f (x (s, p), s) - f (x (s, p_1), s)) d s + \int^r_{r_1} f (x (s, p), s) d s \Vert \le \Vert p - p_1 \Vert + \Vert \int^{r_1}_{r_0} (f (x (s, p)) - f (x (s, p_1))) d s \Vert + \Vert \int^r_{r_1} f (x (s, p)) d s \Vert \le \Vert p - p_1 \Vert + \int^{r_1}_{r_0} \Vert (f (x (s, p)) - f (x (s, p_1))) \Vert d s + \int^r_{r_1} \Vert f (x (s, p)) \Vert d s \le \Vert p - p_1 \Vert + \int^{r_1}_{r_0} L \Vert x (s, p) - x (s, p_1) \Vert d s + \int^r_{r_1} \Vert f (x (s, p)) \Vert d s \le \Vert p - p_1 \Vert + \int^{r_1}_{r_0} L e^{L R} \Vert p - p_1 \Vert d s + \int^r_{r_1} M d s \le \Vert p - p_1 \Vert + L e^{L R} \Vert p - p_1 \Vert R + M (r - r_1)\).

That implies that \(x\) is continuous at \((r_1, p_1)\), obviously.

As \((r_1, p_1) \in J^` \times U^`\) is arbitrary, \(x\) is continuous over \(J^` \times U^`\).

So, the proposition has been proved for \(k = 0\).

Step 4:

Step 4 Strategy: Step 4-1: see that \(\partial_1 x (r, p)\) exists and is continuous; Step 4-2: see that \(\partial_{l + 1} x (r, p)\) exists and is continuous.

Let us prove the proposition for \(k = 1\).

Step 4-1:

For each \((r, p) \in J^` \times U^`\), \(\partial_1 x (r, p) = f (x (r, p), r)\), so, \(\partial_1 x (r, p)\) exists, and as \(x\) is continuous at \((r, p)\) by Step 3 and \(f\) is continuous, \(\partial_1 x\) is continuous at \((r, p)\).

Step 4-2:

Step 4-2 Strategy: Step 4-2-1: define \({g_h}^j_l: J_{r_1} \times B_{p_1, \delta} \to \mathbb{R}, (r, p) \mapsto (x^j (r, p + h e_l) - x^j (r, p)) / h\); Step 4-2-2: define \(s: \mathbb{N} \to \mathbb{R}, n \mapsto \delta (1 / 2)^{n + 1}\) and \(s'_l: \mathbb{N} \to \{: J_{r_1} \times B_{p_1, \delta} \to \mathbb{R}^d\}, n \mapsto {g_{s (n)}}_l\); Step 4-2-3: see that \(s'_l\) is a uniformly Cauchy sequence of maps, by applying the proposition that for any differentiable map from any open interval into any Euclidean \(C^\infty\) manifold with the Euclidean norm, the norm of each point image is upper-bounded by a differentiable map between the non-negative interval with an initial condition if there is a Lipschitz map between the non-negative interval that satisfies certain conditions for \(u: J_{r_1} \to \mathbb{R}^{d^2}, r \mapsto {g_h}_l (r, p) - {g_{h'}}_l (r, p)\), \(v: [0, \infty) \to [0, \infty), r \mapsto 2 c d \rho e^{L R} / (c d A) (e^{c d A r} - 1)\), and \(w: [0, \infty) \to [0, \infty), r \mapsto c d A r + 2 c d \rho e^{L R}\); Step 4-2-4: see that the convergence of \(s'_l\) is \(\partial_{l + 1} x (r, p)\).

Let us see that \(\partial_{l + 1} x (r, p)\) exists and is continuous at each \((r, p) \in J^` \times U^`\).

Step 4-2-1:

Let \(e_l \in \mathbb{R}^d\) be the unit vector of \(\mathbb{R}^d\) in the direction of the \(l\) component.

Let \(h \in B_{0, \delta} \subseteq \mathbb{R}\) be any.

For each \(p \in B_{p_1, \delta}\), \(p + h e_l \in B_{p_1, 2 \delta}\), because \(dist (p + h e_l, p_1) \le dist (p + h e_l, p) + dist (p, p_1) \lt \delta + \delta = 2 \delta\).

Let \({g_h}^j_l: J_{r_1} \times B_{p_1, \delta} \to \mathbb{R}, (r, p) \mapsto (x^j (r, p + h e_l) - x^j (r, p)) / h\).

Let \(g_h: J_{r_1} \times B_{p_1, \delta} \to \mathbb{R}^{d^2}\) be the map whose \((j, l)\) component is \({g_h}^j_l\) above: \(g_h (r, p)\) is regarded to be the \(d \times d\) matrix.

Let \({g_h}_l: J_{r_1} \times B_{p_1, \delta} \to \mathbb{R}^d\) be the map whose \(j\) component is \({g_h}^j_l\) above: \({g_h}_l (r, p)\) is regarded to be the \(d\) vector.

Step 4-2-2:

Let us think of the sequence, \(s: \mathbb{N} \to \mathbb{R}, n \mapsto \delta (1 / 2)^{n + 1}\).

Then, \(s'_l: \mathbb{N} \to \{: J_{r_1} \times B_{p_1, \delta} \to \mathbb{R}^d\}, n \mapsto {g_{s (n)}}_l\) is the sequence of maps.

Step 4-2-3:

Let us see that \(s'_l\) is a uniformly Cauchy sequence of maps.

Let \(A \in \mathbb{R} := Sup (\{\vert \partial_m f^j (p, r) \vert \vert (p, r) \in \overline{V} \times \overline{J_{r_1}}, j, m \in \{1, ..., d\}\})\) or if it is \(0\), let \(A\) be any positive real number, which exists as \(\lt \infty\), by the proposition that the image of any continuous map from any compact topological space to the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum, because \(\overline{V} \times \overline{J_{r_1}}\) is a compact subspace of \(U \times J\) (as is seen in Step 2) and \(\partial_m f^j\) is continuous over \(U \times J\).

Let \(c \in \mathbb{R}\) be any such that for the \(d \times d\) matrix norm induced by vector norms, \(\Vert \bullet \Vert\), and the Frobenius \(d \times d\) matrix norm, \(\Vert \bullet \Vert_F\), \(\Vert \bullet \Vert \lt c \Vert \bullet \Vert_F\): refer to Note for the definition of Frobenius matrix norm.

Let us take \(u: J_{r_1} \to \mathbb{R}^d, r \mapsto {g_h}_l (r, p) - {g_{h'}}_l (r, p)\), \(v: [0, \infty) \to [0, \infty), r \mapsto 2 c d \rho e^{L R} / (c d A) (e^{c d A r} - 1)\), and \(w: [0, \infty) \to [0, \infty), r \mapsto c d A r + 2 c d \rho e^{L R}\).

Let us see that \(u\), \(v\), and \(w\) satisfies the conditions for the proposition that for any differentiable map from any open interval into any Euclidean \(C^\infty\) manifold with the Euclidean norm, the norm of each point image is upper-bounded by a differentiable map between the non-negative interval with an initial condition if there is a Lipschitz map between the non-negative interval that satisfies certain conditions, with \(u\), \(v\) and \(w\) instead of \(f\), \(g\), and \(h\).

\(u\) is differentiable.

\(v\) is differentiable and \(v (0) = \Vert u (r_0) \Vert\), because while \(v (0) = 0\), \(u (r_0) = {g_h}_l (r_0, p) - {g_{h'}}_l (r_0, p) = (x (r_0, p + h e_l) - x (r_0, p)) / h - (x (r_0, p + h' e_l) - x (r_0, p)) / h' = (p + h e_l - p) / h - (p + h' e_l - p) / h' = h e_l / h - h' e_l / h' = e_l - e_l = 0\), so, \(\Vert u (r_0) \Vert = 0\).

\(w\) is a Lipschitz map with \(c d A\), obviously.

Let us see that \(\Vert \partial_1 u (r) \Vert \le w (\Vert u (r) \Vert)\).

\(\partial_1 u (r) = \partial_1 ({g_h}_l (r, p) - {g_{h'}}_l (r, p))\).

\(\partial_1 {g_h}_l (r, p) = (\partial_1 x (r, p + h e_l) - \partial_1 x (r, p)) / h = (f (x (r, p + h e_l), r) - f (x (r, p), r)) / h\).

Let us think of \({y_h}_l (r, p): [0, 1] \to \mathbb{R}^d, s \mapsto f ((1 - s) x (r, p) + s x (r, p + h e_l), r)\), which is valid, because the line segment, \(\overline{x (r, p) x (r, p + h e_l)}\), is contained in \(V\), because \(\Vert x (r, p) - x (r, p_1) \Vert \le e^{L R} \Vert p - p_1 \Vert \lt e^{L R} \delta \lt \epsilon / 2\) while \(k := x (r, p_1) \in K\), which means that \(x (r, p) \in B_{k, \epsilon / 2}\), and \(\Vert x (r, p) - x (r, p + h e_l) \Vert \le e^{L R} \Vert p + h e_l - p \Vert = e^{L R} \vert h \vert \lt e^{L R} \delta \lt \epsilon / 2\), so, \(\Vert x (r, p + h e_l) - k \Vert \le \Vert x (r, p + h e_l) - x (r, p) \Vert + \Vert x (r, p) - k \Vert \lt \epsilon / 2 + \epsilon / 2 = \epsilon\), \(x (r, p), x (r, p + h e_l) \in B_{k, \epsilon}\).

\({y_h}_l (r, p)\) is differentiable, because \(f\) is differentiable and \((1 - s) x (r, p) + s x (r, p + h e_l)\) is differentiable with respect to \(s\).

By the mean-value theorem for any differentiable map from any closed interval into the \(1\)-dimensional Euclidean \(C^\infty\) manifold, there is an \(t_j \in (0, 1)\) such that \({y_h}^j_l (r, p) (1) - {y_h}^j_l (r, p) (0) = \partial_1 {y_h}^j_l (r, p) (t_j)\).

Let \({p_2}_{l, h, j} (r, p) := (1 - t_j) x (r, p) + t_j x (r, p + h e_l)\).

\(f^j (x (r, p + h e_l), r) - f^j (x (r, p), r) = {y_h}^j_l (r, p) (1) - {y_h}^j_l (r, p) (0) = \partial_1 {y_h}^j_l (r, p) (t_j) = \partial_m f^j ({p_2}_{l, h, j} (r, p), r) (x^m (r, p + h e_l) - x^m (r, p)) = \partial_m f^j ({p_2}_{l, h, j} (r, p), r) {g_h}^m_l (r, p) h\).

So, \(\partial_1 {g_h}^j_l (r, p) = \partial_m f^j ({p_2}_{l, h, j} (r, p), r) {g_h}^m_l (r, p)\).

\(\partial_1 ({g_h}^j_l (r, p) - {g_{h'}}^j_l (r, p)) = \partial_m f^j ({p_2}_{l, h, j} (r, p), r) {g_h}^m_l (r, p) - \partial_m f^j ({p_2}_{l, h', j} (r, p), r) {g_{h'}}^m_l (r, p) = \partial_m f^j ({p_2}_{l, h, j} (r, p), r) ({g_h}^m_l (r, p) - {g_{h'}}^m_l (r, p)) + \partial_m f^j ({p_2}_{l, h, j} (r, p), r) {g_{h'}}^m_l (r, p) - \partial_m f^j ({p_2}_{l, h', j} (r, p), r) {g_{h'}}^m_l (r, p) = \partial_m f^j ({p_2}_{l, h, j} (r, p), r) ({g_h}^m_l (r, p) - {g_{h'}}^m_l (r, p)) + (\partial_m f^j ({p_2}_{l, h, j} (r, p), r) - \partial_m f^j ({p_2}_{l, h', j} (r, p), r)) {g_{h'}}^m_l (r, p)\).

So, \(\partial_1 ({g_h}_l (r, p) - {g_{h'}}_l (r, p)) = M_{l, h} (r, p) ({g_h}_l (r, p) - {g_{h'}}_l (r, p)) + (M_{l, h} (r, p) - M_{l, h'} (r, p)) {g_{h'}}_l (r, p)\), where \(M_{l, h} (r, p)\) is the matrix whose \((j, m)\) component is \({M_{l, h}}^j_m (r, p) = \partial_m f^j ({p_2}_{l, h, j} (r, p), r)\): note that \(M_{l, h} (r, p)\) is not exactly the Jacobian at any point, because \({p_2}_{l, h, j} (r, p)\) depends on \(j\), which does not matter.

\(\Vert \partial_1 ({g_h}_l (r, p) - {g_{h'}}_l (r, p)) \Vert = \Vert M_{l, h} (r, p) ({g_h}_l (r, p) - {g_{h'}}_l (r, p)) + (M_{l, h} (r, p) - M_{l, h'} (r, p)) {g_{h'}}_l (r, p) \Vert \le \Vert M_{l, h} (r, p) \Vert \Vert {g_h}_l (r, p) - {g_{h'}}_l (r, p) \Vert + \Vert M_{l, h} (r, p) - M_{l, h'} (r, p) \Vert \Vert {g_{h'}}_l (r, p) \Vert\), where the matrix norms are the norms induced by vector norms, \(\le c \Vert M_{l, h} (r, p) \Vert_F \Vert {g_h}_l (r, p) - {g_{h'}}_l (r, p) \Vert + c \Vert M_{l, h} (r, p) - M_{l, h'} (r, p) \Vert_F \Vert {g_{h'}}_l (r, p) \Vert\), where \(\Vert \bullet \Vert_F\) is the Frobenius matrix norm.

As \(\vert {M_{l, h} (r, p)}^j_m \vert \le A\), \(\Vert M_{l, h} (r, p) \Vert_F \le d A\).

\(\Vert {g_h}_l (r, p) \Vert = \Vert x (r, p + h e_l) - x (r, p) \Vert / \vert h \vert \le e^{L R} \Vert p + h e_l - p \Vert / \vert h \vert = e^{L R} \Vert h e_l \Vert / \vert h \vert = e^{L R}\).

\(\Vert {g_{h'}}_l (r, p) \Vert \le e^{L R}\), likewise.

\(\partial_m f^j\) is continuous over \(U \times J\), so, \(\partial_m f^j\) is uniformly continuous over \(\overline{V} \times \overline{J_{r_1}}\), by the proposition that for any continuous map between any metric spaces with the domain with the induced topology, the restriction of the map on any compact domain is uniformly continuous.

So, for each \(\rho \in \mathbb{R}\) such that \(0 \lt \rho\), there is a \(\lambda \in \mathbb{R}\) such that \(0 \lt \lambda\) and for each \((p, r), (p', r') \in \overline{V} \times \overline{J_{r_1}}\) such that \(\sqrt{\Vert p' - p \Vert^2 + \Vert r' - r \Vert^2} \lt \lambda\), \(\Vert \partial_m f^j (p', r') - \partial_m f^j (p, r) \Vert \lt \rho\).

Let \(\vert h \vert, \vert h' \vert \lt \lambda e^{- L R}\).

Then, \(\Vert {p_2}_{l, h, j} (r, p) - x (r, p) \Vert = \Vert (1 - t_j) x (r, p) + t_j x (r, p + h e_l) - x (r, p) \Vert = \Vert - t_j x (r, p) + t_j x (r, p + h e_l) \Vert = \Vert t_j (x (r, p + h e_l) - x (r, p)) \Vert = t_j \Vert x (r, p + h e_l) - x (r, p) \Vert = t_j \Vert {g_h}_l (r, p) h \Vert \le t_j \vert h \vert e^{L R} \lt \lambda e^{- L R} e^{L R} = \lambda\).

\(\Vert {p_2}_{l, h', j} (r, p) - x (r, p) \Vert \lt \lambda\), likewise.

\(\vert (M_{l, h} (r, p) - M_{l, h'} (r, p))^j_m \vert = \vert \partial_m f^j ({p_2}_{l, h, j} (r, p), r) - \partial_m f^j ({p_2}_{l, h', j} (r, p), r) \vert = \vert \partial_m f^j ({p_2}_{l, h, j} (r, p), r) - \partial_m f^j (x (r, p), r) + \partial_m f^j (x (r, p), r) - \partial_m f^j ({p_2}_{l, h', j} (r, p), r) \vert \le \vert \partial_m f^j ({p_2}_{l, h, j} (r, p), r) - \partial_m f^j (x (r, p), r) \vert + \vert \partial_m f^j ({p_2}_{l, h', j} (r, p), r) - \partial_m f^j (x (r, p), r) \vert \lt \rho + \rho = 2 \rho\), because \(\sqrt{\Vert \Vert {p_2}_{l, h, j} (r, p) - x (r, p) \Vert \Vert^2 + \Vert r - r \Vert^2} \lt \lambda\) and \(\sqrt{\Vert \Vert {p_2}_{l, h', j} (r, p) - x (r, p) \Vert \Vert^2 + \Vert r - r \Vert^2} \lt \lambda\).

So, \(\Vert M_{l, h} (r, p) - M_{l, h'} (r, p) \Vert_F \lt 2 \rho d\).

So, \(\Vert \partial_1 u (r) \Vert \lt c d A \Vert u (r) \Vert + 2 c d \rho e^{L R} = w (\Vert u (r) \Vert)\).

\(\partial_1 v (r) = 2 c d \rho e^{L R} e^{c d A r}\) while \(w (v (r)) = c d A (2 c d \rho e^{L R} / (c d A) (e^{c d A r} - 1)) + 2 c d \rho e^{L R} = 2 c d \rho e^{L R} (e^{c d A r} - 1) + 2 c d \rho e^{L R} = 2 c d \rho e^{L R} e^{c d A r}\), so, \(\partial_1 v (r) = w (v (r))\).

So, \(u, v, w\) satisfies the conditions, so, \(\Vert u (r) \Vert \le v (\vert r - r_0 \vert)\), so, \(\Vert {g_h}_l (r, p) - {g_{h'}}_l (r, p) \Vert \le 2 c d \rho e^{L R} / (c d A) (e^{c d A \vert r - r_0 \vert} - 1) \le 2 c d \rho e^{L R} / (c d A) (e^{c d A R} - 1)\).

So, for any \(\alpha \in \mathbb{R}\) such that \(0 \lt \alpha\), \(\rho\) can be chosen such that \(2 c d \rho e^{L R} / (c d A) (e^{c d A R} - 1) \lt \alpha\), then, \(\lambda\) can be chosen accordingly, and \(N \in \mathbb{N}\) can be chosen such that \(s (N) \lt \lambda e^{- L R}\), then for each \(n, n' \in \mathbb{N}\) such that \(N \lt n, n'\), \(s (n), s (n') \lt \lambda e^{- L R}\), so, \(\Vert s'_l (n) (r, p) - s'_l (n') (r, p) \Vert = \Vert {g_{s (n)}}_l (r, p) - {g_{s (n')}}_l (r, p) \Vert \lt \alpha\).

As \(N\) is chosen independent of \((r, p)\), \(s'_l\) is a uniformly Cauchy sequence.

So, \(s'_l\), converges uniformly to the continuous map, \(z_l: J_{r_1} \times B_{p_1, \delta} \to \mathbb{R}^d\), by the proposition that any uniformly Cauchy sequence of maps from any set into any complete metric space converges uniformly and the proposition that for any uniformly convergent sequence of continuous maps from any topological space into any metric space with the induced topology, the convergence is continuous.

\({z_l}^j\) is indeed the derivative, \(lim_{h \to 0} (x^j (r, p + h e_l) - x^j (r, p)) / h\), because \(\vert (x^j (r, p + h e_l) - x^j (r, p)) / h - {z_l}^j (r, p) \vert = \vert {g_h}^j_l (r, p) - {z_l}^j (r, p) + {g_{s (n)}}^j_l (r, p) - {g_{s (n)}}^j_l (r, p) \vert \le \vert {g_h}^j_l (r, p) - {g_{s (n)}}^j_l (r, p) \vert + \vert {z_l}^j (r, p) - {g_{s (n)}}^j_l (r, p) \vert\), and choosing \(n\) such that \(\vert {z_l}^j (r, p) - {g_{s (n)}}^j_l (r, p) \vert \lt \alpha / 2\) and \(s (n) \lt \lambda e^{- L R}\) for \(\alpha / 2\), \(\lt \vert {g_h}^j_l (r, p) - {g_{s (n)}}^j_l (r, p) \vert + \alpha / 2\), and for each \(\vert h \vert \lt \lambda e^{- L R}\) for \(\alpha / 2\), \(\vert {g_h}^j_l (r, p) - {g_{s (n)}}^j_l (r, p) \vert \lt \alpha / 2\), so, \(\vert (x^j (r, p + h e_l) - x^j (r, p)) / h - {z_l}^j (r, p) \vert \lt \alpha\).

As \(z_l\) exists and is continuous around each \((r_1, p_1) \in J^` \times U^`\), \(z_l\) exists and is continuous over whole \(J^` \times U^`\).

So, the proposition holds for \(k = 1\).

Step 5:

Let us suppose that the proposition holds for \(0 \le k \le k' - 1\) where \(2 \le k'\).

Let us prove that the proposition holds for \(k = k'\).

As \(f\) is \(C^{k' - 1}\), \(x\) is \(C^{k' - 1}\), by the induction hypothesis.

\(\partial_1 x^j = f^j (x (r, p), r)\), which is \(C^{k - 1}\), by the proposition that for any maps between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.

As \(x^j (r, p) = p^j + \int^r_{r_0} f^j (x (s, p), s) d s\), \(\partial_{l + 1} x^j (r, p) = \delta^j_l + \int^r_{r_0} \partial_{l + 1} \widetilde{f}^j (s, p) d s\), where \(\widetilde{f} (s, p)\) is \(f (x (s, p), s)\) regarded as the map from \(J^` \times U^`\), \(C^1\), by the proposition that for any maps between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point, because \(\widetilde{f} = f \circ (x, id) \circ \iota\), where \(\iota: J^` \times U^` \to (J^` \times U^`) \times J^`, (r, p) \mapsto ((r, p), r)\), while the derivation under the integral is possible, because \(\widetilde{f}\) is \(C^1\).

\(\partial_{l + 1} \widetilde{f}^j (s, p) = \partial_m f^j (x (s, p), s) \partial_{l + 1} x^m (s, p)\), by the chain rule.

So, \(\partial_{1 + l} x^j (r, p) = \delta^j_l + \int^r_{r_0} \partial_m f^j (x (s, p), s) \partial_{l + 1} x^m (s, p) d s\).

So, \(\partial_1 \partial_{1 + l} x^j (r, p) = \partial_m f^j (x (r, p), r) \partial_{l + 1} x^m (r, p)\).

Let us think of \((\partial_{1 + l} x, x): J^` \times U^` \to \mathbb{R}^{2 d}\) and \(\partial_1 \partial_{1 + l} x (r, p) = \partial_m f (x (r, p), r) \partial_{l + 1} x^m (r, p)\) and \(\partial_1 x (r, p) = f (x (r, p), r)\) combined, which is an ordinal differential equation with initial condition, \((\partial_{1 + l} x, x) (r_0) = ((\delta^j_l), p)\) with \(f'\) instead of \(f\) such that \(f' ((\partial_{1 + l} x, x), r) = (\partial_m f (x, r) (\partial_{l + 1} x)^m, f (x, r))\).

As \(f'\) is \(C^{k - 1}\), the solution, \((\partial_{1 + l} x, x)\) is \(C^{k - 1}\), by the induction hypothesis.

So, especially, \(\partial_{1 + l} x\) is \(C^{k - 1}\).

As \(\partial_1 x\) and \(\partial_{l + 1} x\) s are \(C^{k - 1}\), \(x\) is \(C^k\).

Step 6:

By the induction principle, the proposition holds for each \(k \in \mathbb{N}\).

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References

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1640: For Metric Space, Open Subset, Compact Subset Contained in Open Subset, There Is Positive Radius of Which Open or Close Ball Around Each Point on Compact Subset Is Contained in Open Subset

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description/proof of that for metric space, open subset, compact subset contained in open subset, there is positive radius of which open or close ball around each point on compact subset is contained in open subset

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Topics

About: metric space
About: topological space

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The table of contents of this article

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

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Starting Context

• The reader knows a definition of metric space.
• The reader knows a definition of topology induced by metric.
• The reader knows a definition of compact subset of topological space.
• The reader knows a definition of open ball around point on metric space.
• The reader knows a definition of closed ball around point on metric space.
• The reader admits the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.
• The reader admits the proposition that for any metric space and any subset, the closure of the subset is compact if and only if each sequence into the subset has a subsequence that converges in the closure of the subset if and only if the closure of the subset is complete and for each positive real number, there is a set of some finite open balls of the-number-radius that covers the closure of the subset.

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Target Context

• The reader will have a description and a proof of the proposition that for any metric space, any open subset, and any compact subset contained in the open subset, there is a positive radius of which the open or closed ball around each point on the compact subset is contained in the open subset.

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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: Structured Description

Here is the rules of Structured Description.

Entities:
\(M\): \(\in \{\text{ the metric spaces }\}\), with the topology induced by the metric
\(U\): \(\in \{\text{ the open subsets of } M\}\)
\(K\): \(\in \{\text{ the compact subsets of } M\}\), such that \(K \subseteq U\)
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Statements:
\(\exists \delta_0 \in \mathbb{R} \text{ such that } 0 \lt \delta_0 (\forall k \in K (B_{k, \delta_0} \subseteq U))\)
\(\land\)
\(\exists \delta'_0 \in \mathbb{R} \text{ such that } 0 \lt \delta'_0 (\forall k \in K (B'_{k, \delta'_0} \subseteq U))\), where \(B'_{k, \delta'_0}\) is the closed ball
//

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2: Proof

Whole Strategy: apply the proposition that for any metric space and any subset, the closure of the subset is compact if and only if each sequence into the subset has a subsequence that converges in the closure of the subset if and only if the closure of the subset is complete and for each positive real number, there is a set of some finite open balls of the-number-radius that covers the closure of the subset; Step 1: suppose that there was no such \(\delta_0\), and find a contradiction; Step 2: take \(\delta'_0\) such that \(\delta'_0 \lt \delta_0\).

Step 1:

Let us suppose that there was no such \(\delta_0\).

For each \(\delta \in \mathbb{R}\) such that \(0 \lt \delta\), there would be a \(k_\delta \in K\) such that \(\lnot B_{k_\delta, \delta} \subseteq U\).

Let us take for each \(j \in \mathbb{N}\), \(\delta_j := 2^{- j}\).

\(s: \mathbb{N} \to K, j \mapsto k_{\delta_j}\) would be a sequence into \(K\).

\(K \subseteq M\) would be a compact subspace, by the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.

There would be a subsequence, \(s \circ f\), where \(f: \mathbb{N} \to \mathbb{N}\), that converges to a \(k \in K\), by the immediate corollary mentioned in the proposition that for any metric space and any subset, the closure of the subset is compact if and only if each sequence into the subset has a subsequence that converges in the closure of the subset if and only if the closure of the subset is complete and for each positive real number, there is a set of some finite open balls of the-number-radius that covers the closure of the subset.

There would be an \(\epsilon \in \mathbb{R}\) such that \(0 \lt \epsilon\) and \(B_{k, \epsilon} \subseteq U\), because \(U\) was open.

There would be an \(N \in \mathbb{N}\) such that for each \(j \in \mathbb{N}\) such that \(N \lt j\), \(s \circ f (j) \in B_{k, \epsilon / 2}\), because \(s \circ f\) converged to \(k\).

There would be an \(N' \in \mathbb{N}\) such that \(N \le N'\) and for each \(j \in \mathbb{N}\) such that \(N' \lt j\), \(\delta_{f (j)} \lt \epsilon / 2\), because \(\delta_{f (j)} = 2^{- f (j)}\).

For each \(j \in \mathbb{N}\), there would be a point, \(m_j \in B_{k_{\delta_{f (j)}}, \delta_{f (j)}} \setminus U\), because \(B_{k_{\delta_{f (j)}}, \delta_{f (j)}}\) was not contained in \(U\).

For each \(j \in \mathbb{N}\) such that \(N' \lt j\), \(dist (k, m_j) \le dist (k, k_{\delta_{f (j)}}) + dist (k_{\delta_{f (j)}}, m_j) \lt \epsilon / 2 + \delta_{f (j)} \lt \epsilon / 2 + \epsilon / 2 = \epsilon\), which would mean that \(m_j \in B_{k, \epsilon} \subseteq U\), a contradiction against \(m_j \in B_{k_{\delta_{f (j)}}, \delta_{f (j)}} \setminus U\).

So, the supposition was wrong, and there is a \(\delta_0\).

Step 2:

There is a \(\delta_0\), by Step 1.

Let \(\delta'_0 \in \mathbb{R}\) be any such that \(0 \lt \delta'_0\) and \(\delta'_0 \lt \delta_0\).

Then, for each \(k \in K\), \(B'_{k, \delta'_0} \subseteq B_{k, \delta_0}\).

So, \(B'_{k, \delta'_0} \subseteq B_{k, \delta_0} \subseteq U\).

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References

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