description/proof of that affine map from affine or convex set spanned by possibly-non-affine-independent set of base points on finite-dimensional real vectors space into finite dimensional real vectors space is continuous w.r.t. canonical topologies
Topics
About: vectors space
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
Starting Context
- The reader knows a definition of affine map from affine set spanned by possibly-non-affine-independent set of base points on real vectors space.
- The reader knows a definition of affine map from convex set spanned by possibly-non-affine-independent set of base points on real vectors space.
- The reader knows a definition of canonical topology for finite-dimensional real vectors space.
- The reader knows a definition of canonical \(C^\infty\) atlas for finite-dimensional real vectors space.
- The reader knows a definition of subspace topology of subset of topological space.
- The reader admits the proposition that any topological spaces map is continuous at any point if the spaces are the subspaces of some \(C^\infty\) manifolds and there are some charts of the manifolds around the point and the point image and a map between the chart open subsets which (the map) is restricted to the original map whose (the chart open subsets map's) coordinates function's restriction is continuous.
- The reader admits the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
Target Context
- The reader will have a description and a proof of the proposition that any affine map from the affine or convex set spanned by any possibly-non-affine-independent set of base points on any finite-dimensional real vectors space into any finite-dimensional real vectors space is continuous with respect to the canonical topologies.
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:
\(V_1\): \(\in \{\text{ the } d_1 \text{ -dimensional real vectors spaces }\}\) with the canonical topology
\(V_2\): \(\in \{\text{ the } d_2 \text{ -dimensional real vectors spaces }\}\) with the canonical topology
\(S_1\): \(\in \{\text{ the affine or convex sets spanned by any possibly non-affine-independent set of base points on } V_1\}\), \(\subseteq V_1\), with the subspace topology
\(f\): \(:S_1 \to V_2\), \(\in \{\text{ the affine maps }\}\)
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Statements:
\(f \in \{\text{ the continuous maps }\}\).
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2: Natural Language Description
For any \(d_1\)-dimensional real vectors space, \(V_1\), with the canonical topology, any \(d_2\)-dimensional real vectors space, \(V_2\), with the canonical topology, the affine or convex set spanned by any possibly non-affine-independent set of base points on \(V_1\), \(S_1\), with the subspace topology, and any affine map, \(f: S_1 \to V_2\), \(f\) is continuous.
3: Proof
There is the affine-independent subset of the base points, \(\{p'_0, ..., p'_k\} \subseteq \{p_0, ..., p_n\}\), with which the affine map is defined (refer to a definition of affine map from affine set spanned by possibly-non-affine-independent set of base points on real vectors space or to a definition of affine map from convex set spanned by possibly-non-affine-independent set of base points on real vectors space), and \(f: \sum_{j \in \{0, ..., k\}} t'^j p'_j \mapsto \sum_{j \in \{0, ..., k\}} t'^j f (p'_j)\).
We are going to apply the proposition that any topological spaces map is continuous at any point if the spaces are the subspaces of some \(C^\infty\) manifolds and there are some charts of the manifolds around the point and the point image and a map between the chart open subsets which (the map) is restricted to the original map whose (the chart open subsets map's) coordinates function's restriction is continuous.
There are a basis of \(V_1\), \(\{p'_1 - p'_0, ..., p'_k - p'_0, v_{k + 1}, ..., v_{d_1}\}\), and the canonical chart, \((V_1 \subseteq V_1, \phi)\). Now, \(V_1\) is the \(C^\infty\) manifold, of which \(S_1\) is the topological subspace.
For each \(p \in S_1\), \(p = \sum_{j \in \{0, ..., k\}} t'^j p'_j = \sum_{j \in \{0, ..., k\}} t'^j (p'_j - p'_0) + \sum_{j \in \{0, ..., k\}} t'^j p'_0 = \sum_{j \in \{1, ..., k\}} t'^j (p'_j - p'_0) + p'_0\). \(\phi (p) = (t'^1 + {p'_0}^1, ..., t'^k + {p'_0}^k, {p'_0}^{k + 1}, ..., {p'_0}^{d_1})\), where \(\phi (p'_0) = ({p'_0}^1, ..., {p'_0}^{d_1})\).
There are a basis of \(V_2\), \(\{w_1, ..., w_{d_2}\}\), and the canonical chart, \((V_2 \subseteq V_2, \psi)\). Now, \(V_2\) is the \(C^\infty\) manifold.
\(\psi (\sum_{j \in \{0, ..., k\}} t'^j f (p'_j)) = (\sum_{j \in \{0, ..., k\}} t'^j q_j^1, ..., \sum_{j \in \{0, ..., k\}} t'^j q_j^{d_2})\), where \(\psi (f (p'_j)) = (q_j^1, ..., q_j^{d_2})\).
Let us think of the extension of \(f\), \(f': V_1 \to V_2, \sum_{j \in \{1, ..., k\}} t'^j (p'_j - p'_0) + p'_0 + \sum_{j \in \{k + 1, ..., d_1\}} t'^j v_j \mapsto \sum_{j \in \{0, ..., k\}} t'^j f (p'_j)\), where \(t'^j\) is extended to \((- \infty, \infty)\). \(f\) is indeed the restriction of \(f'\) on the domain, \(S_1\).
The coordinates function of \(f'\) with respect to the charts is, \(\psi \circ f \circ {\phi}^{-1}: \mathbb{R}^{d_1} \to \mathbb{R}^{d_2}, (t'^1 + {p'_0}^1, ..., t'^k + {p'_0}^k, t'^{k + 1} + {p'_0}^{k + 1}, ..., t'^{d_1} + {p'_0}^{d_1}) \mapsto (\sum_{j \in \{0, ..., k\}} t'^j q_j^1, ..., \sum_{j \in \{0, ..., k\}} t'^j {q_j}^{d_2})\); taking \(t''^j := t'^j + {p'_0}^j\), it is \((t''^1, ..., t''^k, t''^{k + 1}, ..., t''^{d_1}) \mapsto (\sum_{j \in \{0, ..., k\}} (t''^j - {p'_0}^j) {q_j}^1, ..., \sum_{j \in \{0, ..., k\}} (t''^j - {p'_0}^j) {q_j}^{d_2})\), which is continuous as it is linear. Then, also the restriction of the coordinates function, \(\psi \circ f \circ {\phi}^{-1} \vert_{\phi (V_1 \cap S_1)}\), is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
So, by the proposition that any topological spaces map is continuous at any point if the spaces are the subspaces of some \(C^\infty\) manifolds and there are some charts of the manifolds around the point and the point image and a map between the chart open subsets which (the map) is restricted to the original map whose (the chart open subsets map's) coordinates function's restriction is continuous, \(f\) is continuous.
