A description/proof of that linear map between Euclidean topological spaces is continuous
Topics
About: topological space
The table of contents of this article
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 \(C^\infty\) 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.
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.
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: Description
For any Euclidean topological spaces, \(\mathbb{R}^{d1}, \mathbb{R}^{d2}\), any linear map, \(f: \mathbb{R}^{d1} \rightarrow \mathbb{R}^{d2}\) is continuous.
2: Proof
There are the canonical Euclidean manifolds, \(\mathbb{R}^{d1} \subseteq \mathbb{R}^{d1}, \mathbb{R}^{d2} \subseteq \mathbb{R}^{d2}\), which are \(C^\infty\) manifolds, whose subspaces the Euclidean topological spaces are. There are the canonical charts, \(\lt\mathbb{R}^{di}, \phi'_i\gt\) such that \(\phi'_i: \mathbb{R}^{di} \rightarrow \mathbb{R}^{di}\) is the identity map.
There is the map, \(f': \mathbb{R}^{d1} \rightarrow \mathbb{R}^{d2}\), between the Euclidean manifolds such that \(f' = f\). The coordinates functions, \(\phi'_2 \circ f' \circ {\phi'_1}^{-1}\), 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 \(C^\infty\) 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 \(p \in \mathbb{R}^{d1}\).
