definition of convergence of series on 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 series.
Target Context
- The reader will have a definition of convergence of series on 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:
\( J\): \(\subseteq \mathbb{N}\), such that \(J \neq \emptyset\)
\( R\): \(\in \{\text{ the rings }\} \cap \{\text{ the metric spaces }\}\)
\( s\): \(\in \{\text{ the sequences }\}\), such that \(Dom (s) = J\) and \(Ran (s) \subseteq R\)
\( \widetilde{s}\): \(= \sum_{j \in \{1, 2, ...\}} s_j\)
\( \widetilde{s}'\): \(: J \to R, j \mapsto \sum_{l \in \{J_1, ... , j\}} s (l)\), \(\in \{\text{ the sequences }\}\)
\(*r\): \(\in R\)
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Conditions:
\(\widetilde{s}' \text{ converges to } r\)
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2: Note
\(s\) is a sequence, \(\widetilde{s}\) is the series that corresponds to \(s\), and \(\widetilde{s}'\) is the sequence that corresponds to \(\widetilde{s}\), and any convergence of \(\widetilde{s}\) is the convergence of \(\widetilde{s}'\), the definition says.
\(\widetilde{s}\) does not necessarily have a convergence, because \(\widetilde{s}'\) does not necessarily have a convergence.
Any convergence, \(r\), is inevitably unique, because any convergence of any sequence on metric space is unique, as is mentioned in Note for the definition of convergence of sequence on metric space.
