description/proof of that linear map between vectors metric spaces induced by norms is continuous iff it is bounded
Topics
About: vectors space
About: metric 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 linear map.
- The reader knows a definition of continuous map.
- The reader knows a definition of bounded map between normed vectors spaces.
Target Context
- The reader will have a description and a proof of the proposition that any linear map between any vectors metric spaces induced by any norms is continuous if and only if it is bounded.
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 the metric induced by any norm, \(\Vert \bullet \Vert_1\)
\(V_2\): \(\in \{\text{ the } F \text{ vectors spaces }\}\) with the metric induced by any norm, \(\Vert \bullet \Vert_2\)
\(*f\): \(: V_1 \to V_2\), \(\in \{\text{ the linear maps }\}\)
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Statements:
\(f \in \{\text{ the continuous maps }\}\)
\(\iff\)
\(f \in \{\text{ the bounded maps }\}\)
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2: Proof
Whole Strategy: Step 1: suppose that \(f\) is bounded; Step 2: for each \(v_1 \in V_1\), take any neighborhood of \(f (v_1)\), \(N_{f (v_1)} \subseteq V_2\), take any open ball around \(f (v_1)\), \(B_{f (v_1), \epsilon} \subseteq V_2\), such that \(B_{f (v_1), \epsilon} \subseteq N_{f (v_1)}\), take the open ball around \(v_1\), \(B_{v_1, \epsilon / c} \subseteq V_1\), where \(c\) is any constant for being bounded, and see that \(f (B_{v_1, \epsilon / c}) \subseteq B_{f (v_1), \epsilon}\); Step 3: suppose that \(f\) is continuous; Step 4: take the open ball around \(0 \in V_2\), \(B_{0, 1} \subseteq V_2\), take an open ball around \(0 \in V_1\), \(B_{0, \delta} \subseteq V_1\), such that \(f (B_{0, \delta}) \subseteq B_{0, 1}\); Step 5: for each \(v_1 \in V_1\) such that \(v_1 \neq 0\), evaluate \(\Vert f (v_1) \Vert\), using \(v_1 = (\delta / 2) / (\delta / 2) \vert v_1 \vert / \vert v_1 \vert v_1\).
Step 1:
Let us suppose that \(f\) is bounded.
Step 2:
Let \(v_1 \in V_1\) be any.
Let \(N_{f (v_1)} \subseteq V_2\) be any neighborhood of \(f (v_1)\).
There is an open ball around \(f (v_1)\), \(B_{f (v_1), \epsilon} \subseteq V_2\), such that \(B_{f (v_1), \epsilon} \subseteq N_{f (v_1)}\).
As \(f\) is bounded, there is a constant, \(c \in \mathbb{R}\), such that for each \(v'_1 \in V_1\), \(\Vert f (v'_1) \Vert \le c \Vert v'_1 \Vert\).
Let us take the open ball around \(v_1\), \(B_{v_1, \epsilon / c} \subseteq V_1\).
For each \(v'_1 \in B_{v_1, \epsilon / c}\), \(f (v'_1) = f (v'_1 - v_1 + v_1) = f (v'_1 - v_1) + f (v_1)\), so, \(f (v'_1) - f (v_1) = f (v'_1 - v_1)\).
\(\Vert f (v'_1) - f (v_1) \Vert = \Vert f (v'_1 - v_1)\Vert \le c \Vert v'_1 - v_1 \Vert \lt c \epsilon / c = \epsilon\).
That means that \(f (B_{v_1, \epsilon / c}) \subseteq B_{f (v_1), \epsilon} \subseteq N_{f (v_1)}\).
That means that \(f\) is continuous at \(v_1\).
As \(v_1 \in V_1\) is arbitrary, \(f\) is continuous.
Step 3:
Let us suppose that \(f\) is continuous.
Step 4:
As \(f\) is linear, \(f (0) = 0\).
Let us take the open ball around \(f (0) = 0 \in V_2\), \(B_{0, 1} \subseteq V_2\).
As \(f\) is continuous, there is an open ball around \(0 \in V_1\), \(B_{0, \delta} \subseteq V_1\), such that \(f (B_{0, \delta}) \subseteq B_{0, 1}\).
Step 5:
For each \(v_1 \in V_1\) such that \(v_1 \neq 0\), \(\Vert f (v_1) \Vert = \Vert f ((\delta / 2) / (\delta / 2) \Vert v_1 \Vert / \Vert v_1 \Vert v_1) \Vert = \Vert \Vert v_1 \Vert / (\delta / 2) f ((\delta / 2) / \Vert v_1 \Vert v_1) \Vert = \Vert v_1 \Vert / (\delta / 2) \Vert f ((\delta / 2) / \Vert v_1 \Vert v_1) \Vert\), but \((\delta / 2) / \Vert v_1 \Vert v_1 \in B_{0, \delta}\), so, \(f ((\delta / 2) / \Vert v_1 \Vert v_1) \in B_{0, 1}\) and \(\Vert f (v_1) \Vert \lt \Vert v_1 \Vert / (\delta / 2) 1 \le 1 / (\delta / 2) \Vert v_1 \Vert\).
So, \(c := 1 / (\delta / 2)\), which does not depend on \(v_1\), makes \(\Vert f (v_1) \Vert \le c \Vert v_1 \Vert\).
So, \(f\) is bounded.
