335: Linear Map Between Euclidean Topological Spaces Is Continuous

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A description/proof of that linear map between Euclidean topological spaces is continuous

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Topics

About: topological 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 Euclidean topological space.
• The reader knows a definition of continuous map.
• The reader admits the proposition that any topological spaces map is continuous at any point if there are super $C^{\mathrm{\infty }}$ manifolds of the topological spaces and a map between them which (the map) restricts to the original map on a chart open set of the domain manifold around the point whose (the manifolds map's) coordinates functions are continuous.

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Target Context

• The reader will have a description and a proof of the proposition that any linear map between any Euclidean topological spaces is continuous.

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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 Euclidean topological spaces, ${\mathbb{R}}^{d1},{\mathbb{R}}^{d2}$, any linear map, $f:{\mathbb{R}}^{d1}\to {\mathbb{R}}^{d2}$ is continuous.

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2: Proof

There are the canonical Euclidean manifolds, ${\mathbb{R}}^{d1}\subseteq {\mathbb{R}}^{d1},{\mathbb{R}}^{d2}\subseteq {\mathbb{R}}^{d2}$, which are $C^{\mathrm{\infty }}$ manifolds, whose subspaces the Euclidean topological spaces are. There are the canonical charts, $<{\mathbb{R}}^{di},{\varphi }_i^{\prime }>$ such that ${\varphi }_i^{\prime }:{\mathbb{R}}^{di}\to {\mathbb{R}}^{di}$ is the identity map.

There is the map, $f^{\prime }:{\mathbb{R}}^{d1}\to {\mathbb{R}}^{d2}$, between the Euclidean manifolds such that $f^{\prime }=f$. The coordinates functions, ${\varphi }_2^{\prime }\circ f^{\prime }\circ {{\varphi }_1^{\prime }}^{-1}$, are practically the same with $f$ and are obviously continuous as $f$ is linear.

By the proposition that any topological spaces map is continuous at any point if there are super $C^{\mathrm{\infty }}$ manifolds of the topological spaces and a map between them which (the map) restricts to the original map on a chart open set of the domain manifold around the point whose (the manifolds map's) coordinates functions are continuous, $f$ is continuous at any point $p\in {\mathbb{R}}^{d1}$.

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References

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