344: Finite Dimensional Real Vectors Space Topology Defined Based on Coordinates Space Does Not Depend on Choice of Basis

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A description/proof of that finite dimensional real vectors space topology defined based on coordinates space does not depend on choice of basis

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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: Description
• 2: Proof

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Starting Context

• The reader knows a definition of vectors space.
• The reader knows a definition of topology.
• The reader admits the proposition that for any map between $C^{\mathrm{\infty }}$ manifolds, its continuousness in the topological sense equals its continuousness in the norm sense for the coordinates functions.

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Target Context

• The reader will have a description and a proof of the proposition that for any finite dimensional real vectors space, the topology defined by the Euclidean topology of the coordinates space based on any basis does not depend on the choice of basis.

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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: Description

For any finite dimensional real vectors space, $V$, any basis, $(b_1,b_2,...,b_d)$, and the topology, $T$, defined by the Euclidean topology of the coordinates space based on the basis, the topology does not depend on the choice of basis.

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2: Proof

The map from V to the coordinates space is a bijection, $f:V\to {\mathbb{R}}^d$. For any another basis, $(b_1^{\prime },b_2^{\prime },...,b_d^{\prime })$, the map from $V$ to the coordinates space is a bijection, $f^{\prime }:V\to {\mathbb{R}}^d=\varphi \circ f$ where $\varphi$ is the bijective coordinates transition map, $\varphi :{\mathbb{R}}^d\to {\mathbb{R}}^d$. The coordinate spaces are given the Euclidean topologies and the Euclidean atlases, and $V$ is given the topologies, $T$ and $T^{\prime }$, based on the Euclidean topologies, which means that any subset of $V$ is open if and only if the corresponding subset of the coordinates space is open.

By the proposition that for any map between $C^{\mathrm{\infty }}$ manifolds, its continuousness in the topological sense equals its continuousness in the norm sense for the coordinates functions, $\varphi$ is a homeomorphism, because its and its inverse's coordinates functions are continuous being linear. So, any open set in one of the coordinates spaces corresponds to an open set in the other, so, whatever open on $T$ is open on $T^{\prime }$ and vice versa,

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References

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