LF-space | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-27

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/LF-space


00:02:19 Properties
00:02:35 Examples



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SUMMARY
=======
In mathematics, an LF-space is a topological vector space V that is a locally convex inductive limit of a countable inductive system 
  
    
      
        (
        
          V
          
            n
          
        
        ,
        
          i
          
            n
            m
          
        
        )
      
    
    {\displaystyle (V_{n},i_{nm})}
   of Fréchet spaces. This means that V is a direct limit of the system 
  
    
      
        (
        
          V
          
            n
          
        
        ,
        
          i
          
            n
            m
          
        
        )
      
    
    {\displaystyle (V_{n},i_{nm})}
   in the category of locally convex topological vector spaces and each 
  
    
      
        
          V
          
            n
          
        
      
    
    {\displaystyle V_{n}}
   is a Fréchet space.
Some authors restrict the term LF-space to mean that V is a strict locally convex inductive limit, which means that the topology induced on 
  
    
      
        
          V
          
            n
          
        
      
    
    {\displaystyle V_{n}}
   by 
  
    
      
        
          V
          
            n
            +
            1
          
        
      
    
    {\displaystyle V_{n+1}}
   is identical to the original topology on 
  
    
      
        
          V
          
            n
          
        
      
    
    {\displaystyle V_{n}}
  .The topology on V can be described by specifying that an absolutely convex subset U is a neighborhood of 0 if and only if 
  
    
      
        U
        ∩
        
          V
          
            n
          
        
      
    
    {\displaystyle U\cap V_{n}}
   is an absolutely convex neighborhood of 0 in 
  
    
      
        
          V
          
            n
          
        
      
    
    {\displaystyle V_{n}}
   for every n.