Complete topological vector space | Wikipedia audio article
Channel: wikipedia tts
Date: 28/03/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Complete_metric_space


00:00:53 1 Examples
00:04:39 2 Some theorems
00:08:13 3 Completion
00:12:04 4 Topologically complete spaces
00:13:35 5 Alternatives and generalizations
00:14:57 6 See also



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SUMMARY
=======
In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.
Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. 
  
    
      
        
          
            2
          
        
      
    
    {\displaystyle {\sqrt {2}}}
   is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it. (See the examples below.)  It is always possible to "fill all the holes", leading to the completion of a given space, as explained below.