description/proof of that multiplication on finite-product of real matrices topological spaces is continuous
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of %ring name% matrices space.
- The reader knows a definition of \(m \times n\) real matrices topological space.
- The reader knows a definition of product topological space.
- The reader knows a definition of continuous, topological spaces map.
- The reader knows a definition of finite-product \(c^\infty\) manifold with boundary.
- The reader admits the proposition that any topological spaces map is continuous at any point if the spaces are the subspaces of some \(C^\infty\) manifolds and there are some charts of the manifolds around the point and the point image and a map between the chart open subsets which (the map) is restricted to the original map whose (the chart open subsets map's) coordinates function's restriction is continuous.
Target Context
- The reader will have a description and a proof of the proposition that the multiplication on any multiplication-possible finite-product of real matrices 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:
\(M_{n_1 \times n_2} (\mathbb{R})\): \(= \text{ the real matrices topological space }\)
\(M_{n_2 \times n_3} (\mathbb{R})\): \(= \text{ the real matrices topological space }\)
\(...\):
\(M_{n_m \times n_{m + 1}} (\mathbb{R})\): \(= \text{ the real matrices topological space }\)
\(m\): \(: M_{n_1 \times n_2} (\mathbb{R}) \times ... \times M_{n_m \times n_{m + 1}} (\mathbb{R}) \to M_{n_1 \times n_{m + 1}} (\mathbb{R})\), \(= \text{ the multiplication }\)
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Statements:
\(m \in \{\text{ the continuous maps }\}\)
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2: Proof
Whole Strategy: Step 1: take the homeomorphisms, \(h_{n_j, n_l}: M_{n_j, n_l} (\mathbb{R}) \to \mathbb{R}^{n_j n_l}\), and see that \(m = {h_{n_1, n_{m + 1}}}^{- 1} \circ h_{n_1, n_{m + 1}} \circ m \circ ({h_{n_1, n_2}}^{- 1}, ..., {h_{n_m, n_{m + 1}}}^{- 1}) \circ (h_{n_1, n_2}, ..., h_{n_m, n_{m + 1}})\); Step 2: see that \(h_{n_1, n_{m + 1}} \circ m \circ ({h_{n_1, n_2}}^{- 1}, ..., {h_{n_m, n_{m + 1}}}^{- 1})\) is continuous; Step 3: conclude the proposition.
Step 1:
Let us take the homeomorphisms, for each \(n_j, n_l \in \mathbb{N} \setminus \{0\}\), \(h_{n_j, n_l}: M_{n_j, n_l} (\mathbb{R}) \to \mathbb{R}^{n_j n_l}\), by which the topology of \(M_{n_j, n_l} (\mathbb{R})\) is defined.
\(m = {h_{n_1, n_{m + 1}}}^{- 1} \circ h_{n_1, n_{m + 1}} \circ m \circ ({h_{n_1, n_2}}^{- 1}, ..., {h_{n_m, n_{m + 1}}}^{- 1}) \circ (h_{n_1, n_2}, ..., h_{n_m, n_{m + 1}})\), because for each \((M_1, ..., M_m) \in M_{n_1 \times n_2} (\mathbb{R}) \times ... \times M_{n_m \times n_{m + 1}} (\mathbb{R})\), \({h_{n_1, n_{m + 1}}}^{- 1} \circ h_{n_1, n_{m + 1}} \circ m \circ ({h_{n_1, n_2}}^{- 1}, ..., {h_{n_m, n_{m + 1}}}^{- 1}) \circ (h_{n_1, n_2}, ..., h_{n_m, n_{m + 1}}) (M_1, ..., M_m) = {h_{n_1, n_{m + 1}}}^{- 1} \circ h_{n_1, n_{m + 1}} \circ m (({h_{n_1, n_2}}^{- 1}, ..., {h_{n_m, n_{m + 1}}}^{- 1}) (h_{n_1, n_2} (M_1), ..., h_{n_m, n_{m + 1}} (M_m))) = {h_{n_1, n_{m + 1}}}^{- 1} \circ h_{n_1, n_{m + 1}} \circ m ({h_{n_1, n_2}}^{- 1} (h_{n_1, n_2} (M_1)), ..., {h_{n_m, n_{m + 1}}}^{- 1} (h_{n_m, n_{m + 1}} (M_m))) = {h_{n_1, n_{m + 1}}}^{- 1} \circ h_{n_1, n_{m + 1}} \circ m (M_1, ..., M_m) = m (M_1, ..., M_m)\).
Step 2:
Let us see that \(h_{n_1, n_{m + 1}} \circ m \circ ({h_{n_1, n_2}}^{- 1}, ..., {h_{n_m, n_{m + 1}}}^{- 1})\) is continuous.
\(\mathbb{R}^{n_1 n_2} \times ... \times \mathbb{R}^{n_m n_{m + 1}}\) is a topological subspace of the finite-product \(C^{\infty}\) manifold, \(\mathbb{R}^{n_1 n_2} \times ... \times \mathbb{R}^{n_m n_{m + 1}}\), and \(\mathbb{R}^{n_1 n_{m + 1}}\) is a topological subspace of the \(C^{\infty}\) manifold, \(\mathbb{R}^{n_1 n_{m + 1}}\), and by the standard charts for \(\mathbb{R}^{n_1 n_2} \times ... \times \mathbb{R}^{n_m n_{m + 1}}\) and \(\mathbb{R}^{n_1 n_{m + 1}}\), each component of each \((v_1, ..., v_m) \in \mathbb{R}^{n_1 n_2} \times ... \times \mathbb{R}^{n_m n_{m + 1}}\), which is a component of a \(v_j \in \mathbb{R}^{n_j n_{j + 1}}\), becomes a component of \({h_{n_j, n_{j + 1}}}^{- 1} (v_j)\), each component of \(m ({h_{n_1, n_2}}^{- 1} (v_1), ..., {h_{n_m, n_{m + 1}}}^{- 1} (v_m))\) is a polynomial of the components of \({h_{n_j, n_{j + 1}}}^{- 1} (v_j)\) s, which is a polynomial of the components of \(v_j\) s, and those components of \(m ({h_{n_1, n_2}}^{- 1} (v_1), ..., {h_{n_m, n_{m + 1}}}^{- 1} (v_m))\) become the components of \(h_{n_1, n_{m + 1}} \circ m \circ ({h_{n_1, n_2}}^{- 1}, ..., {h_{n_m, n_{m + 1}}}^{- 1}) (v_1, ..., v_m)\), so, the components of the components function of \(h_{n_1, n_{m + 1}} \circ m \circ ({h_{n_1, n_2}}^{- 1}, ..., {h_{n_m, n_{m + 1}}}^{- 1})\) are some polynomials, which are continuous.
So, \(h_{n_1, n_{m + 1}} \circ m \circ ({h_{n_1, n_2}}^{- 1}, ..., {h_{n_m, n_{m + 1}}}^{- 1})\) is continuous, by the proposition that any topological spaces map is continuous at any point if the spaces are the subspaces of some \(C^\infty\) manifolds and there are some charts of the manifolds around the point and the point image and a map between the chart open subsets which (the map) is restricted to the original map whose (the chart open subsets map's) coordinates function's restriction is continuous.
Step 3:
\((h_{n_1, n_2}, ..., h_{n_m, n_{m + 1}})\) is continuous, by the proposition that the product map of any finite number of continuous maps is continuous by the product topologies.
\({h_{n_1, n_{m + 1}}}^{- 1}\) is continuous, because \(h_{n_1, n_{m + 1}}\) is a homeomorphism.
So, \(m = {h_{n_1, n_{m + 1}}}^{- 1} \circ h_{n_1, n_{m + 1}} \circ m \circ ({h_{n_1, n_2}}^{- 1}, ..., {h_{n_m, n_{m + 1}}}^{- 1}) \circ (h_{n_1, n_2}, ..., h_{n_m, n_{m + 1}})\) 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.
