definition of uniformly convergent sequence of maps from set into metric space
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of metric space.
- The reader knows a definition of map.
Target Context
- The reader will have a definition of uniformly convergent sequence of maps from set into metric space.
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:
\( S\): \(\in \{\text{ the sets }\}\)
\( M\): \(\in \{\text{ the metric spaces }\}\)
\( J\): \(\subseteq \mathbb{N}\)
\(*s\): \(: J \to \{g: S \to M\}\)
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Conditions:
\(\exists f: S \to M (\forall \epsilon \in \mathbb{R} \text{ such that } 0 \lt \epsilon (\exists N \in J (\forall j \in J \text{ such that } N \lt j (\forall p \in S (dist (s (j) (p), f (p)) \lt \epsilon)))))\)
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2: Note
The point is that \(N\) does not depend on \(p\).
\(M\) needs to be a metric space instead of a general topological space, because while the point of this concept is to take the same-sized open ball around each \(f (p)\), "same-sized"-ness does not make sense for a general topological space.
