description/proof of that linear map between complex Euclidean topological spaces is continuous
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of complex Euclidean topological space.
- The reader knows a definition of linear map.
- The reader knows a definition of continuous map.
- The reader admits the proposition that any complex vectors space can be regarded to be the canonical real vectors space.
- 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 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.
Target Context
- The reader will have a description and a proof of the proposition that any linear map between any complex 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: Structured Description
Here is the rules of Structured Description.
Entities:
\(d_1\): \(\in \mathbb{N} \setminus \{0\}\)
\(d_2\): \(\in \mathbb{N} \setminus \{0\}\)
\(\mathbb{C}^{d_1}\): \(= \text{ the complex Euclidean topological space }\), also as the complex Euclidean vectors space
\(\mathbb{C}^{d_2}\): \(= \text{ the complex Euclidean topological space }\), also as the complex Euclidean vectors space
\(f\): \(: \mathbb{C}^{d_1} \to \mathbb{C}^{d_2}\), \(\in \{\text{ the linear maps }\}\)
Statements:
\(f \in \{\text{ the continuous maps }\}\)
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2: Proof
Whole Strategy: Step 1: take \(f_1: \mathbb{R}^{2 d_1} \to \mathbb{C}^{d_1}\) and \(f_2: \mathbb{R}^{2 d_2} \to \mathbb{C}^{d_2}\) as the canonical homeomorphisms; Step 2: see that \(f = f_2 \circ {f_2}^{-1} \circ f \circ f_1 \circ {f_1}^{-1}\); Step 3: see that \(f_2\), \({f_2}^{-1} \circ f \circ f_1\), and \({f_1}^{-1}\) are continuous.
Step 1:
Let us take the canonical homeomorphisms, \(f_1: \mathbb{R}^{2 d_1} \to \mathbb{C}^{d_1}\) and \(f_2: \mathbb{R}^{2 d_2} \to \mathbb{C}^{d_2}\): they are some homeomorphisms because the topologies of \(\mathbb{C}^{d_1}\) and \(\mathbb{C}^{d_2}\) have been defined to make them homeomorphisms.
Step 2:
\(f = f_2 \circ {f_2}^{-1} \circ f \circ f_1 \circ {f_1}^{-1}\).
Step 3:
\(f_2\) is continuous, because it is a homeomorphism.
\({f_1}^{-1}\) is a homeomorphism, because \(f_1\) is a homeomorphism, and is continuous.
Let us see that \(f_1\) is a 'vectors spaces - linear morphisms' isomorphism with \(\mathbb{C}^{d_1}\) regarded as a \(\mathbb{R}\) vectors space: refer to the proposition that any complex vectors space can be regarded to be the canonical real vectors space.
For each \(r, r' \in \mathbb{R}^{2 d_1}\) and \(s, s' \in \mathbb{R}\), \(f_1 (s r + s' r') = f_1 (s (r^1, r^2, ..., r^{2 (d_1 - 1) + 1}, r^{2 d_1}) + s' (r'^1, r'^2, ..., r'^{2 (d_1 - 1) + 1}, r'^{2 d_1})) = f_1 ((s r^1 + s' r'^1, s r^2 + s' r'^2, ..., s r^{2 (d_1 - 1) + 1} + s' r'^{2 (d_1 - 1) + 1}, s r^{2 d_1} + s' r'^{2 d_1})) = (s r^1 + s' r'^1 + (s r^2 + s' r'^2) i, ..., s r^{2 (d_2 - 1) + 1} + s' r'^{2 (d_1 - 1) + 1} + (s r^{2 d_1} + s' r'^{2 d_1}) i) = s (r^1 + r^2 i, ..., r^{2 (d_1 - 1) + 1} + r^{2 d_1} i) + s' (r'^1 + r'^2 i, ..., r'^{2 (d_1 - 1) + 1} + r'^{2 d_1} i) = s f_1 (r) + s' f_1 (r')\), so, \(f_1\) is \(\mathbb{R}\)-linear.
By the proposition that any bijective linear map between any vectors spaces is a 'vectors spaces - linear morphisms' isomorphism, \(f_2\) is a 'vectors spaces - linear morphisms' isomorphism.
Likewise, \(f_2\) is a 'vectors spaces - linear morphisms' isomorphism with \(\mathbb{C}^{d_2}\) regarded as a \(\mathbb{R}\) vectors space.
So, \({f_2}^{-1}\) is \(\mathbb{R}\)-linear.
\(f\) is \(\mathbb{R}\)-linear with \(\mathbb{C}^{d_1}\) and \(\mathbb{C}^{d_2}\) regarded as the \(\mathbb{R}\) vectors spaces, because it is \(\mathbb{C}\)-linear.
So, \({f_2}^{-1} \circ f \circ f_1: \mathbb{R}^{2 d_1} \to \mathbb{R}^{2 d_2}\) is an \(\mathbb{R}\)-linear map between Euclidean vectors spaces, 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.
By the proposition that any linear map between any Euclidean topological spaces is continuous, \({f_2}^{-1} \circ f \circ f_1\) is continuous.
So, \(f = f_2 \circ {f_2}^{-1} \circ f \circ f_1 \circ {f_1}^{-1}\) 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.
