description/proof of that for map between normed vectors spaces s.t. image norm divided by argument norm converges to 0 when argument norm nears 0, image norm of map plus nonzero linear map divided by argument norm does not do so
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
Target Context
- The reader will have a description and a proof of the proposition that for any map between any normed vectors spaces such that the image norm divided by the argument norm converges to 0 when the argument norm nears 0, the image norm of the map plus any nonzero linear map 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:
\(V_1\): \(\in \{\text{ the normed vectors spaces }\}\)
\(V_2\): \(\in \{\text{ the normed vectors spaces }\}\)
\(f_1\): \(: V_1 \to V_2\)
\(f_2\): \(: V_1 \to V_2\), \(\in \{\text{ the linear maps }\}\)
//
Statements:
(
\(lim_{\Vert v \Vert \to 0} \Vert f_1 (v) \Vert / \Vert v \Vert = 0\)
\(\land\)
\(\exists v_0 \in V_1 (f_2 (v_0) \neq 0)\)
)
\(\implies\)
\(\Vert (f_1 + f_2) (v) \Vert / \Vert v \Vert\) does not converge to 0 when \(\Vert v \in V_1 \Vert\) nears 0
//
2: Natural Language Description
For any normed vectors spaces, \(V_1, V_2\), any map, \(f_1: V_1 \to V_2\) such that \(lim_{\Vert v \Vert \to 0} \Vert f_1 (v) \Vert / \Vert v \Vert = 0\), and any nonzero linear map, \(f_2: V_1 \to V_2\), \(\Vert (f_1 + f_2) (v) \Vert / \Vert v \Vert\) does not converge to 0 when \(\Vert v \in V_1 \Vert\) nears 0.
3: Proof
Whole Strategy: Step 1: suppose that \(lim_{\Vert v \Vert \to 0} \Vert f_1 (v) \Vert / \Vert v \Vert = 0\) and \(\exists v_0 \in V_1 (f_2 (v_0) \neq 0)\) and suppose that \(lim_{\Vert v \Vert \to 0} \Vert (f_1 + f_2) (v) \Vert / \Vert v \Vert = 0\), and find a contradiction.
Let us define \(f_3: V_1 \to V_2 := f_1 + f_2\).
Let us suppose that \(lim_{\Vert v \Vert \to 0} \Vert f_3 (v) \Vert / \Vert v \Vert = 0\).
\(f_3 - f_1 = f_2\). \(\Vert (f_3 - f_1) (v) \Vert / \Vert v \Vert = \Vert f_3 (v) - f_1 (v) \Vert / \Vert v \Vert \le (\Vert f_3 (v) \Vert + \Vert f_1 (v) \Vert) / \Vert v \Vert = \Vert f_3 (v) \Vert / \Vert v \Vert + \Vert f_1 (v) \Vert / \Vert v \Vert\), which would converge to \(0\) when \(\Vert v \Vert\) neared \(0\), because the both terms would do so, so, \(\Vert (f_3 - f_1) (v) \Vert / \Vert v \Vert\) would converge to \(0\). That is a contradiction, because \(\Vert f_2 (v) \Vert / \Vert v \Vert\) does not converge to \(0\) when \(\Vert v \Vert\) nears \(0\), by 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.
