345: 'Real Vectors Spaces-Linear Morphisms' Isomorphism Between Topological Spaces with Coordinates Topologies Is Homeomorphic

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A description/proof of that 'real vectors spaces-linear morphisms' isomorphism between topological spaces with coordinates topologies is homeomorphic

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Topics

About: vectors space
About: topological space
About: map

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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 topological space.
• The reader admits 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.
• 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 2 finite dimensional vectors space topological spaces with the coordinates based topologies, any 'real vectors spaces-linear morphisms' isomorphism between them is a homeomorphism.

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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 spaces, $V_1$ and $V_2$, made topological spaces, $T_1$ and $T_2$, with the topologies defined by the Euclidean topologies of the coordinates spaces (it has been proved that the topologies do not depend on choices of bases), any 'real vectors spaces-linear morphisms' isomorphism, $f:T_1\to T_2$, is a homeomorphism.

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2: Proof

$T_1$ and $T_2$ are canonically $C^{\mathrm{\infty }}$ manifolds with their coordinate spaces as chart open sets. As $f$ is linear, its coordinates function from the $T_1$ coordinates space to the $T_2$ coordinates space is linear, so, continuous in the norm sense. So, 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, $f$ is continuous in the topological sense. Likewise, $f^{-1}$ is continuous.

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References

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