561: Canonical C∞ Atlas for Finite-Dimensional Real Vectors Space

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definition of canonical $C^{\mathrm{\infty }}$ atlas for finite-dimensional real vectors space

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Topics

About: vectors space
About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

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

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Starting Context

• The reader knows a definition of %field name% vectors space.
• The reader knows a definition of canonical topology for finite-dimensional real vectors space.
• The reader knows a definition of $C^{\mathrm{\infty }}$ manifold.

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Target Context

• The reader will have a definition of canonical $C^{\mathrm{\infty }}$ atlas for finite-dimensional real vectors space.

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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:
$V$: $\in \{\text{ the }d\text{ -dimensional real vectors spaces }\}$
$\{b_1,...,b_d\}$: $\subseteq V$, $\in \{\text{ the bases of }V\}$
${\mathbb{R}}^d$: $=\text{ the Euclidean topological space }$
$f$: $:V\to {\mathbb{R}}^d$, $v\mapsto (v^1,...,v^d)$ such that $v=v^j{b}_j$
$O$: $=\{U\subseteq V|f(U)\in \text{ the topology of }{\mathbb{R}}^d\}$
$(V,f)$: $\in \{\text{ the charts for }V\}$
$\ast A$: $=\text{ the maximal }{C}^{\mathrm{\infty }}\text{ atlas compatible with }(V,f)$
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Conditions:
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$A$ does not depend on the choice of $\{b_1,...,b_d\}$, because with another $\{b_1^{\prime },...,b_d^{\prime }\}$ and the corresponding $f^{\prime }:V\to {\mathbb{R}}^d$, the charts transition maps, $f\circ f^{\prime -1}:{\mathbb{R}}^d\to {\mathbb{R}}^d$ and $f^{\prime }\circ f^{-1}:{\mathbb{R}}^d\to {\mathbb{R}}^d$ are $C^{\mathrm{\infty }}$: $f\circ f^{\prime -1}$ is linear with a non-singular matrix and $f^{\prime }\circ f^{-1}$ is its inverse.

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2: Natural Language Description

For any $d$-dimensional real vectors space, $V$, any basis, $\{b_1,...,b_d\}\subseteq V$, the Euclidean topological space, ${\mathbb{R}}^d$, the map, $f:V\to {\mathbb{R}}^d$, $v\mapsto (v^1,...,v^d)$ such that $v=v^j{b}_j$, and the chart, $(V,f)$, the maximal $C^{\mathrm{\infty }}$ atlas compatible with $(V,f)$

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3: Note

As $V$ with $O$ is obviously a Hausdorff, 2nd-countable, locally topological Euclidean topological space, $V$ with $O$ and $A$ is a $C^{\mathrm{\infty }}$ manifold.

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References

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