---
date: 2022-06-05 23:24
modification date: Sunday 5th June 2022 23:24:14
title: "spherically complete"
aliases: [spherically complete]
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- Tags: 
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- Refs:
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# spherically complete

Needed to prove Hahn-Banach for ultrametric spaces:
![](attachments/Pasted%20image%2020220605232436.png)

Consequences:

- Every shrinking map (not just contracting maps) has a unique fixed point.
- If $Y\injects X$ with $Y$ spherically complete then for every $x\in X$ there exists a (non-unique) best approximation in $Y$.