description/proof of that motion is injective
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 motion.
- The reader knows a definition of injection.
Target Context
- The reader will have a description and a proof of the proposition that any motion is injective.
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:
\( F_1\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\( F_2\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\(V_1\): \(\in \{\text{ the normed } F_1 \text{ vectors spaces }\}\)
\(V_2\): \(\in \{\text{ the normed } F_2 \text{ vectors spaces }\}\)
\(f\): \(: V_1 \to V_2\), \(\in \{\text{ the motions }\}\)
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Statements:
\(f \in \{\text{ the injections }\}\)
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2: Natural Language Description
For any \(F_1 \in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure, any \(F_2 \in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure, any normed \(F_1\) vectors space, \(V_1\), any normed \(F_2\) vectors space, \(V_2\), and any motion, \(f: V_1 \to V_2\), \(f\) is an injection.
3: Proof
Whole Strategy: Step 1: choose any distinct elements, \(v, v' \in V_1\); Step 2: suppose that \(f (v) = f (v')\) and find a contradiction.
Step 1:
Let us choose any elements, \(v, v' \in V_1\), such that \(v \neq v'\).
Step 2:
Let us suppose that \(f (v) = f (v')\).
\(\Vert v - v' \Vert = \Vert f (v) - f (v') \Vert = \Vert 0 \Vert = 0\), which implies that \(v - v' = 0\), which implies that \(v = v'\), a contradiction.
So, \(f (v) \neq f (v')\).
