definition of normed vectors spaces map continuous 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.
- The reader knows a definition of limit of normed vectors spaces map at point.
Target Context
- The reader will have a definition of normed vectors spaces map continuous 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
\( V_1\): \(\in \{\text{ the normed vectors spaces over } F_1\}\)
\( V_2\): \(\in \{\text{ the normed vectors spaces over } F_2\}\)
\( p\): \(\in V_1\)
\(*f\): \(: V_1 \to V_2\)
//
Conditions:
\(\exists lim_p f \in V_2 \land lim_p f = f (p)\).
//
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 \(F_2\), \(V_2\), and any point, \(p \in V_1\), any map, \(f: V_1 \to V_2\), such that the limit at \(p\), \(lim_p f\), exists and \(lim_p f = f (p)\)
3: Note
This definition is equivalent with this definition:for any sequence, \(p_1, p_2, ...\), where \(p_j \in V_1\), that converges to \(p\), the sequence, \(f (p_1), f (p_2), ...\), converges to \(f (p)\).
Let us confirm that fact.
Let us suppose that \(f\) is continuous at \(p\) by the definition of this article. For any \(\epsilon\), there is a \(\delta\) such that for each \(p'\) such that \(\Vert p' - p \Vert \lt \delta\), \(\Vert f (p') - f (p) \Vert \lt \epsilon\). For any sequence that converges to \(p\), \(p_1, p_2, ...\), there is a \(k\) such that for each \(k \lt j\), \(\Vert p_j - p \Vert \lt \delta\). Then, for each \(k \lt j\), \(\Vert f (p_j) - f (p) \Vert \lt \epsilon\), which means that \(f (p_1), f (p_2), ...\) converges to \(f (p)\).
For the other direction, let us think of the contraposition. Let us suppose that \(f\) is not continuous at \(p\). That means that for an \(\epsilon\), there is no \(\delta\) such that for each \(p' \in V_1\) such that \(\Vert p' - p \Vert \lt \delta\), \(\Vert f (p') - f (p) \Vert \lt \epsilon\), which implies that for each \(\delta\), there is a \(p'\) such that \(\Vert p' - p \Vert \lt \delta\) but \(\epsilon \le \Vert f (p') - f (p) \Vert\). Let us take \(\delta_1 = 1 / 1, \delta_2 = 1 / 2, ...\) and \(p'_1, p'_2, ...\) as \(\Vert p'_j - p \Vert \lt \delta_j\) and \(\epsilon \le \Vert f (p'_j) - f (p) \Vert\). Then, \(p'_1, p'_2, ...\) converges to \(p\), but \(f (p'_1), f (p'_2), ...\) does not converge to \(f (p)\), which is the negation of 'for any sequence, \(p_1, p_2, ...\), where \(p_j \in V_1\), that converges to \(p\), the sequence, \(f (p_1), f (p_2), ...\), converges to \(f (p)\)'. So, the contraposition is true, and so, the other direction is true.
