definition of limit of normed vectors spaces map at point
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: Note
Starting Context
- The reader knows a definition of normed vectors space.
- The reader knows a definition of map.
Target Context
- The reader will have a definition of limit of normed vectors spaces map at point.
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
\( V_1\): \(\in \{\text{ the normed vectors spaces over } F_1\}\)
\( V_2\): \(\in \{\text{ the normed vectors spaces over } F_2\}\)
\( f\): \(: V_1 \to V_2\)
\( p\): \(\in V_1\)
\(*lim_p f\): \(\in V_2\)
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Conditions:
\(\forall \epsilon \in \mathbb{R} \text{ such that } 0 \lt \epsilon (\exists \delta \in \mathbb{R} \text{ such that } 0 \lt \delta (\forall p' \in V_1 \text{ such that } \Vert p' - p \Vert \lt \delta (\Vert f (p') - lim_p f \Vert \lt \epsilon)))\).
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2: Natural Language Description
For \(\mathbb{R}\) or \(\mathbb{C}\) with the canonical field structure, \(F_1\), \(\mathbb{R}\) or \(\mathbb{C}\) with the canonical field structure, \(F_2\), any normed vectors space over \(F_1\), \(V_1\), any normed vectors space over \(F_2\), \(V_2\), any map, \(f: V_1 \to V_2\), and any point, \(p \in V_1\), \(lim_p f \in V_2\) such that for each \(\epsilon \in \mathbb{R}\) such that \(0 \lt \epsilon\), there is a \(\delta \in \mathbb{R}\) such that \(0 \lt \delta\) and for each \(p' \in V_1\) such that \(\Vert p' - p \Vert \lt \delta\), \(\Vert f (p') - lim_p f \Vert \lt \epsilon\)
3: Note
\(lim_p f\) does not necessarily exist.
There cannot be multiple limits, because supposing that a \(lim_p f\) exist, for each \(l \in V_2\) such that \(l \neq lim_p f\), for each \(p' \in V_1\), \(\Vert l - lim_p f \Vert = \Vert l - f (p') + f (p') - lim_p f \Vert \le \Vert l - f (p') \Vert + \Vert f (p') - lim_p f \Vert\), which implies that \(\Vert l - lim_p f \Vert - \Vert f (p') - lim_p f \Vert \le \Vert l - f (p') \Vert\), but \(0 \lt \Vert l - lim_p f \Vert\), and for a \(\delta\), for each \(p' \in V_1\) such that \(\Vert p' - p \Vert \lt \delta\), \(\Vert f (p') - lim_p f \Vert \lt 1 / 2 \Vert l - lim_p f \Vert\), which implies that \(1 / 2 \Vert l - lim_p f \Vert \lt \Vert l - f (p') \Vert\), and for \(\epsilon = 1 / 2 \Vert l - lim_p f \Vert\), for whatever \(\delta'\), there is a \(p'\) such that \(\Vert p' - p \Vert \lt min (\delta, \delta')\) and \(\epsilon \lt \Vert l - f (p') \Vert\), which implies that \(l\) is not any limit.
\(lim_p f\) does not necessarily equal \(f (p)\).
If \(lim_p f\) exists and equals \(f (p)\), \(f\) is continuous at \(p\).
