description/proof of that for map between normed vectors spaces, map is continuous iff map is continuous as map between topological spaces induced by metrics induced by norms
Topics
About: vectors space
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of metric induced by norm on real or complex vectors space.
- The reader knows a definition of topology induced by metric.
- The reader knows a definition of continuous, normed vectors spaces map.
- The reader knows a definition of continuous, topological spaces map.
Target Context
- The reader will have a description and a proof of the proposition that for any map between any normed vectors spaces, the map is continuous if and only if the map is continuous as the map between the topological spaces induced by the metrics induced by the norms.
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
\(V_1\): \(\in \{\text{ the } F \text{ vectors spaces }\}\), with any norm, with the topology induced by the metric induced by the norm
\(V_2\): \(\in \{\text{ the } F \text{ vectors spaces }\}\), with any norm, with the topology induced by the metric induced by the norm
\(f\): \(: V_1 \to V_2\)
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Statements:
\(f \in \{\text{ the continuous, normed vectors spaces maps }\}\)
\(\iff\)
\(f \in \{\text{ the continuous, topological spaces maps }\}\)
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2: Proof
Whole Strategy: Step 1: suppose that \(f\) is a continuous, normed vectors spaces map, and see that \(f\) is a continuous, topological spaces map; Step 2: suppose that \(f\) is a continuous, topological spaces map, and see that \(f\) is a continuous, normed vectors spaces map.
Step 1:
Let us suppose that \(f\) is a continuous, normed vectors spaces map.
Let \(t \in T_1\) be any.
Let \(U_{f (t)} \subseteq T_2\) be any open neighborhood of \(f (t)\).
There is an \(\epsilon\)-'open ball' around \(f (t)\), \(B_{f (t), \epsilon} \subseteq T_2\), such that \(0 \lt \epsilon\) and \(B_{f (t), \epsilon} \subseteq U_{f (t)}\).
There is a \(\delta \in \mathbb{R}\) such that \(0 \lt \delta\) and \(f (B_{t, \delta}) \subseteq B_{f (t), \epsilon}\).
\(B_{t, \delta}\) is an open neighborhood of \(t\), by Note for the definition of topology induced by metric.
As \(f (B_{t, \delta}) \subseteq B_{f (t), \epsilon} \subseteq U_{f (t)}\), \(f\) is continuous at \(t\) as the topological spaces map.
As \(t\) is arbitrary, \(f\) is a continuous, topological spaces map.
Step 2:
Let us suppose that \(f\) is a continuous, topological spaces map.
Let \(t \in T_1\) be any.
Let \(B_{f (t), \epsilon}\) be the \(\epsilon\)-'open ball' around \(f (t)\), where \(\epsilon \in \mathbb{R}\) be any such that \(0 \lt \epsilon\).
\(B_{f (t), \epsilon}\) is an open neighborhood of \(f (t)\), by Note for the definition of topology induced by metric.
So, there is an open neighborhood of \(t\), \(U_t \subseteq T_1\), such that \(f (U_t) \subseteq B_{f (t), \epsilon}\).
There is a \(\delta \in \mathbb{R}\) such that \(0 \lt \delta\) and \(B_{t, \delta} \subseteq U_t\).
\(f (B_{t, \delta}) \subseteq f (U_t) \subseteq B_{f (t), \epsilon}\).
So, \(f\) is continuous at \(t\) as the normed vectors spaces map.
As \(t\) is arbitrary, \(f\) is a continuous, normed vectors spaces map.
