description/proof of that for nonzero linear map between normed vectors spaces, image norm divided by argument norm does not converge to 0 when argument norm nears 0
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
Starting Context
- The reader knows a definition of normed vectors space.
- The reader knows a definition of linear map.
Target Context
- The reader will have a description and a proof of the proposition that for any nonzero linear map between any normed vectors spaces, the image norm divided by the argument norm does not converge to 0 when the argument norm nears 0.
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:
\(V_1\): \(\in \{\text{ the normed vectors spaces }\}\)
\(V_2\): \(\in \{\text{ the normed vectors spaces }\}\)
\(f\): \(: V_1 \to V_2\), \(\in \{\text{ the linear maps }\}\)
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Statements:
\(\exists v_0 \in V_1 (f (v_0) \neq 0)\)
\(\implies\)
\(\Vert f (v) \Vert / \Vert v \Vert\) does not converge to 0 when \(\Vert v \in V_1 \Vert\) nears 0
//
2: Natural Language Description
For any normed vectors spaces, \(V_1, V_2\), and any nonzero linear map, \(f: V_1 \to V_2\), \(\Vert f (v) \Vert / \Vert v \Vert\) does not converge to 0 when \(\Vert v \in V_1 \Vert\) nears 0.
3: Proof
Whole Strategy: Step 1: choose any vector, \(v_0 \in V_1\), such that \(f (v_0) \neq 0\), take \(r v_0\) and make \(r\) approach 0, and see that it does not converge to 0.
Step 1:
As \(f\) is nonzero, there is a vector, \(v_0 \in V_1\), such that \(f (v_0) \neq 0\).
Let us take \(v = r v_0\). \(\Vert f (v) \Vert / \Vert v \Vert = \Vert f (r v_0) \Vert / \Vert r v_0 \Vert = \Vert r f (v_0) \Vert / \Vert r v_0 \Vert = \vert r \vert \Vert f (v_0) \Vert / (\vert r \vert \Vert v_0 \Vert) = \Vert f (v_0) \Vert / \Vert v_0 \Vert \neq 0\), which does not near \(0\) when \(\Vert r v_0 \Vert\) nears \(0\). Of course, it may be \(0\) for a vector, \(v_1 \in V_1\), but it is not said to converge to \(0\) unless it nears \(0\) in any way in which \(\Vert v \Vert\) nears \(0\).
