description/proof of that for linear injection between finite-dimensional real or complex vectors spaces with canonical topologies, its codomain restriction is homeomorphism
Topics
About: vectors space
About: topological space
The table of contents of this article
Starting Context
- 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 knows a definition of subspace topology of subset of topological space.
- The reader knows a definition of injection.
- The reader knows a definition of homeomorphism.
- The reader admits the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
- The reader admits the proposition that the range of any linear map between any vectors spaces is a vectors subspace of the codomain.
- The reader admits the proposition that any bijective linear map between any vectors spaces is a 'vectors spaces - linear morphisms' isomorphism.
- The reader admits the proposition that any linear map between any finite-dimensional real or complex vectors spaces with the canonical topologies is continuous.
- The reader admits the proposition that for any finite-dimensional real or complex vectors space with the canonical topology, the canonical topology of any vectors subspace is the subspace topology.
Target Context
- The reader will have a description and a proof of the proposition that for any linear injection between any finite-dimensional real or complex vectors spaces with the canonical topologies, its codomain restriction is a homeomorphism.
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
\(f\): \(: V_1 \to V_2\), \(\in \{\text{ the linear injections }\}\)
\(f'\): \(: V_1 \to f (V_1) \subseteq V_2\), as the restriction of \(f\)
//
Statements:
\(f' \in \{\text{ the homeomorphisms }\}\)
//
2: Proof
Whole Strategy: Step 1: see that \(f\) is continuous, and that \(f'\) is continuous; Step 2: see that \(f'\) is a 'vectors spaces - linear morphisms' isomorphism; Step 3: see that \(f'^{-1}\) is continuous from the canonical topological space of \(f (V_1)\); Step 4: see that the canonical topological space of \(f (V_1)\) equals the topological subspace of \(V_2\).
Step 1:
\(f\) is continuous, by the proposition that any linear map between any finite-dimensional real or complex vectors spaces with the canonical topologies is continuous.
\(f'\) is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
Step 2:
\(f (V_1)\) is a vectors subspace of \(V_2\), by the proposition that the range of any linear map between any vectors spaces is a vectors subspace of the codomain.
As \(f'\) is a bijective linear map, \(f'\) is a 'vectors spaces - linear morphisms' isomorphism, by the proposition that any bijective linear map between any vectors spaces is a 'vectors spaces - linear morphisms' isomorphism.
Step 3:
So, \(f'^{-1}\) is linear.
\(f'^{-1}\) is continuous from the canonical topological space of \(f (V_1)\), by the proposition that any linear map between any finite-dimensional real or complex vectors spaces with the canonical topologies is continuous, but the issue is that we are talking about \(f'^{-1}\) from the topological subspace of \(V_2\), not from the canonical topological space.
Step 4:
But by the proposition that for any finite-dimensional real or complex vectors space with the canonical topology, the canonical topology of any vectors subspace is the subspace topology, in fact, \(f (V_1)\) as the topological subspace of \(V_2\) equals the canonical topological space.
So, \(f'^{-1}\) is continuous from the topological subspace of \(V_2\).
So, \(f'\) is a homeomorphism.
