description/proof of that linear map between finite-dimensional real or complex vectors spaces with canonical topologies is continuous
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of linear map.
- The reader knows a definition of canonical topology for finite-dimensional real vectors space.
- The reader knows a definition of canonical topology for finite-dimensional complex vectors space.
- The reader admits the proposition that for any finite-dimensional vectors space and any basis, the vectors space is 'vectors spaces - linear morphisms' isomorphic to the components vectors space with respect to the basis.
- The reader admits the proposition that for any linear map between any modules and any linear map from any supermodule of the codomain of the 1st map into any module, the composition of the 2nd map after the 1st map is linear.
- The reader admits the proposition that any linear map between any Euclidean topological spaces is continuous.
- The reader admits the proposition that for any maps between any arbitrary subspaces of any topological spaces continuous at any corresponding points, the composition is continuous at the point.
- The reader admits the proposition that any linear map between any complex Euclidean topological spaces is continuous.
Target Context
- The reader will have a description and a proof of the proposition that any linear map between any finite-dimensional real or complex vectors spaces with the canonical topologies 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: Structured Description
Here is the rules of Structured Description.
Entities:
\(F\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\(d_1\): \(\in \mathbb{N} \setminus \{0\}\)
\(d_2\): \(\in \mathbb{N} \setminus \{0\}\)
\(V_1\): \(\in \{\text{ the } d_1 \text{ -dimensional } F \text{ vectors spaces }\}\), with the canonical topology
\(V_2\): \(\in \{\text{ the } d_2 \text{ -dimensional } F \text{ vectors spaces }\}\), with the canonical topology
\(f\): \(: V_1 \to V_2\), \(\in \{\text{ the linear maps }\}\)
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Statements:
\(f \in \{\text{ the continuous maps }\}\)
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2: Note
This proposition depends on that \(V_1\) and \(V_2\) are finite-dimensional: compare with the proposition that any linear map between any vectors metric spaces induced by any norms is continuous if and only if it is bounded.
3: Proof
Whole Strategy: Step 1: take any basis for \(V_1\) and any basis for \(V_2\) and the 'vectors spaces - linear morphisms' isomorphisms onto the components vectors spaces with respect to the bases, \(f_1, f_2\); Step 2: see that \(f = {f_2}^{-1} \circ f_2 \circ f \circ {f_1}^{-1} \circ f_1\); Step 3: see that \({f_2}^{-1}\), \(f_2 \circ f \circ {f_1}^{-1}\), and \(f_1\) are continuous.
Step 1:
Let us take any basis for \(V_1\), \(B_1\), and any basis for \(V_2\), \(B_2\).
Let \(f_1: V_1 \to F^{d_1}\) be the 'vectors spaces - linear morphisms' isomorphism onto the components vectors space with respect to \(B_1\): refer to the proposition that for any finite-dimensional vectors space and any basis, the vectors space is 'vectors spaces - linear morphisms' isomorphic to the components vectors space with respect to the basis.
Let \(f_2: V_2 \to F^{d_2}\) be the 'vectors spaces - linear morphisms' isomorphism onto the components vectors space with respect to \(B_2\), likewise.
Step 2:
\(f = {f_2}^{-1} \circ f_2 \circ f \circ {f_1}^{-1} \circ f_1\).
Step 3:
\(f_2\) is a homeomorphism, because the topology of \(V_2\) has been defined to make \(f_2\) homeomorphic.
So, \({f_2}^{-1}\) is a homeomorphism, so, continuous.
\(f_1\) is a homeomorphism likewise, and is continuous.
\(f_2 \circ f \circ {f_1}^{-1}\) is linear, by the proposition that for any linear map between any modules and any linear map from any supermodule of the codomain of the 1st map into any module, the composition of the 2nd map after the 1st map is linear.
So, it is continuous, by the proposition that any linear map between any Euclidean topological spaces is continuous or the proposition that any linear map between any complex Euclidean topological spaces is continuous.
So, \(f = {f_2}^{-1} \circ (f_2 \circ f \circ {f_1}^{-1}) \circ f_1\) is continuous, by the proposition that for any maps between any arbitrary subspaces of any topological spaces continuous at any corresponding points, the composition is continuous at the point.
