description/proof of for finite-dimensional real or complex vectors space with canonical topology and open neighborhood of origin, how to get open neighborhood of origin difference of whose 2 points is contained in 1st neighborhood of origin
Topics
About: vectors space
About: topological space
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of for any finite-dimensional real or complex vectors space with the canonical topology and any open neighborhood of the origin, how to get an open neighborhood of the origin difference of whose any 2 points is contained in the 1st neighborhood of the origin.
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: Structured Description
Here is the rules of Structured Description.
Entities:
\(F\): \(\in \{ \mathbb{R}, \mathbb{C} \}\), with the canonical field structure
\(V\): \(\in \{\text{ the } d \text{ -dimensional } F \text{ vectors spaces }\}\), with the canonical topology
\(U_0\): \(\in \{\text{ the open neighborhoods of } 0\}\)
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Statements:
\(\exists B_{0, \epsilon} \subseteq U_0 (\forall v_1, v_2 \in B_{0, \epsilon / 2} (v_1 - v_2 \in B_{0, \epsilon} \subseteq U_0))\)
\(\land\)
\(\exists C_{0, \epsilon} \subseteq U_0 (\forall v_1, v_2 \in C_{0, \epsilon / 2} (v_1 - v_2 \in C_{0, \epsilon} \subseteq U_0))\), where \(C_{0, \epsilon}\) and \(C_{0, \epsilon / 2}\) are the open cubes
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2: Proof
Whole Strategy: equip \(V\) with the metric that corresponds to the metric on \(\mathbb{R}^d\) or \(\mathbb{R}^{2 d}\); Step 1: take any \(B_{0, \epsilon}\) such that \(B_{0, \epsilon} \subseteq U_0\); Step 2: see that \(\forall v_1, v_2 \in B_{0, \epsilon / 2} (v_1 - v_2 \in B_{0, \epsilon} \subseteq U_0)\); Step 3: take any \(C_{0, \epsilon}\) such that \(C_{0, \epsilon} \subseteq U_0\); Step 4: see that \(\forall v_1, v_2 \in C_{0, \epsilon / 2} (v_1 - v_2 \in C_{0, \epsilon} \subseteq U_0)\).
Step 1:
Let \(B = \{b_1, ..., b_d\}\) be any basis for \(V\) and \(f: V \to F^d, v = v^j b_j \mapsto (v^1, ..., v^d)\).
\(V\) has the topology homeomorphic to the Euclidean or complex Euclidean topological space, \(\mathbb{R}^d \text{ or } \mathbb{C}^d\), by \(f\). Furthermore, when \(F = \mathbb{C}\), \(\mathbb{C}^d\) is homeomorphic to \(\mathbb{R}^{2 d}\) by \(g: \mathbb{C}^d \to \mathbb{R}^{2 d}, (v^1, ..., v^d) \mapsto (Re (v^1), Im (v^1), ..., Re (v^d), Im (v^d))\), and \(V\) has the topology homeomorphic to the Euclidean topological space, \(\mathbb{R}^{2 d}\), by \(g \circ f\).
The Euclidean topology is induced by the Euclidean metric on \(\mathbb{R}^d \text{ or } \mathbb{R}^{2 d}\).
Let \(V\) be equipped with the corresponding metric: \(dist (v_1 = {v_1}^j b_j, v_2 = {v_2}^j b_j) = dist (({v_1}^1, ..., {v_1}^d), ({v_2}^1, ..., {v_2}^d))\) or \(dist (v_1 = {v_1}^j b_j, v_2 = {v_2}^j b_j) = dist ((Re ({v_1}^1), Im ({v_1}^1), ..., Re ({v_1}^d), Im ({v_1}^d)), (Re ({v_2}^1), Im ({v_2}^1), ..., Re ({v_2}^d), Im ({v_2}^d))\).
Obviously, the topology of \(V\) is induced by the metric.
Let us take any open ball around \(0\), \(B_{0, \epsilon}\), such that \(B_{0, \epsilon} \subseteq U_0\), which exists because the topology is induced by the metric.
Step 2:
Let us take \(B_{0, \epsilon / 2}\).
Let \(v_1, v_2 \in B_{0, \epsilon / 2}\) be any.
\(dist (v_1 - v_2, 0) = dist (v_1, v_2)\), because when \(F = \mathbb{R}\), \(dist (v_1 - v_2, 0) = dist (({v_1}^1 - {v_2}^1, ..., {v_1}^d - {v_2}^d), (0, ..., 0)) = \sqrt{\sum_{j \in \{1, ...., d\}} ({v_1}^1 - {v_2}^1 - 0)^2} = \sqrt{\sum_{j \in \{1, ...., d\}} ({v_1}^1 - {v_2}^1)^2} = dist (({v_1}^1, ..., {v_1}^d), ({v_2}^1, ..., {v_2}^d)) = dist (v_1, v_2)\); when \(F = \mathbb{C}\), \(dist (v_1 - v_2, 0) = dist ((Re ({v_1}^1 - {v_2}^1), Im ({v_1}^1 - {v_2}^1), ..., Re ({v_1}^d - {v_2}^d), Im ({v_1}^d - {v_2}^d)), (0, ..., 0)) = \sqrt{\sum_{j \in \{1, ...., d\}} ((Re ({v_1}^j) - Re ({v_2}^j) - 0)^2 + (Im ({v_1}^j) - Im ({v_2}^j) - 0)^2}) = \sqrt{\sum_{j \in \{1, ...., d\}} ((Re ({v_1}^j) - Re ({v_2}^j))^2 + (Im ({v_1}^j) - Im ({v_2}^j))^2}) = dist ((Re ({v_1}^1), Im ({v_1}^1), ..., Re ({v_1}^d), Im ({v_1}^d)), (Re ({v_2}^1), Im ({v_2}^1), ..., Re ({v_2}^d), Im ({v_2}^d))) = dist (v_1, v_2)\).
\(dist (v_1, v_2) \le dist (v_1, 0) + dist (0, v_2) \lt \epsilon / 2 + \epsilon / 2 = \epsilon\).
So, \(dist (v_1 - v_2, 0) \lt \epsilon\).
So, \(v_1 - v_2 \in B_{0, \epsilon} \subseteq U_0\).
Step 3:
Let us take any open cube around \(0\), \(C_{0, \epsilon}\), such that \(C_{0, \epsilon} \subseteq U_0\), which exists because the topology is induced by the metric: while there is an open ball around \(0\), \(B_{0, \delta}\), such that \(B_{0, \delta} \subseteq U_0\), there is \(\epsilon := \sqrt{\delta^2 / d} \text{ or } \sqrt{\delta^2 / (2 d)}\) such that \(d \epsilon^2 = \delta^2 \text{ or } 2 d \epsilon^2 = \delta^2\), which means that \(C_{0, \epsilon} \subseteq B_{0, \delta} \subseteq U_0\).
Step 4:
Let us take \(C_{0, \epsilon / 2}\).
Let \(v_1 = {v_1}^j b_j, v_2 = {v_2}^j b_j \in C_{0, \epsilon / 2}\) be any, which means nothing but \(\vert {v_1}^j \vert \lt \epsilon / 2 \text{ or } \vert Re ({v_1}^j) \vert \lt \epsilon / 2 \text{ and } \vert Im ({v_1}^j) \vert \lt \epsilon / 2\) and \(\vert {v_2}^j \vert \lt \epsilon / 2 \text{ or } \vert Re ({v_2}^j) \vert \lt \epsilon / 2 \text{ and } \vert Im ({v_2}^j) \vert \lt \epsilon / 2\).
\(v_1 - v_2 = {v_1}^j b_j - {v_2}^j b_j = ({v_1}^j - {v_2}^j) b_j\).
But \(\vert {v_1}^j - {v_2}^j \vert \le \vert {v_1}^j \vert + \vert {v_2}^j \vert \lt \epsilon / 2 + \epsilon / 2 = \epsilon\) or \(\vert Re ({v_1}^j) - Re ({v_2}^j) \vert \le \vert Re ({v_1}^j) \vert + \vert Re ({v_2}^j) \vert \lt \epsilon / 2 + \epsilon / 2 = \epsilon\) and \(\vert Im ({v_1}^j) - Im ({v_2}^j) \vert \le \vert Im ({v_1}^j) \vert + \vert Im ({v_2}^j) \vert \lt \epsilon / 2 + \epsilon / 2 = \epsilon\).
That means that \(v_1 - v_2 \in C_{0, \epsilon} \subseteq U_0\).
