Fredholm operator | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-05-06

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Fredholm_operator


00:03:26 1 Properties
00:06:53 2 Examples
00:12:25 3 Applications
00:12:55 4 B-Fredholm operators



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.8232106106214432
Voice name: en-AU-Wavenet-B


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm.
A Fredholm operator is a bounded linear operator T : X → Y between two Banach spaces with finite-dimensional kernel 
  
    
      
        ker
        ⁡
        T
      
    
    {\displaystyle \ker T}
   and (algebraic) cokernel 
  
    
      
        
          c
          o
          k
          e
          r
        
        
        T
        =
        Y
        
          /
        
        
          r
          a
          n
        
        
        T
      
    
    {\displaystyle \mathrm {coker} \,T=Y/\mathrm {ran} \,T}
  , and with closed range 
  
    
      
        
          r
          a
          n
        
        
        T
      
    
    {\displaystyle \mathrm {ran} \,T}
  . The last condition is actually redundant.Equivalently, an operator T : X → Y is Fredholm if it is invertible modulo compact operators, i.e., if there exists a bounded linear operator

  
    
      
        S
        :
        Y
        →
        X
      
    
    {\displaystyle S:Y\to X}
  such that

  
    
      
        
          
            I
            d
          
          
            X
          
        
        −
        S
        T
        
        
          and
        
        
        
          
            I
            d
          
          
            Y
          
        
        −
        T
        S
      
    
    {\displaystyle \mathrm {Id} _{X}-ST\quad {\text{and}}\quad \mathrm {Id} _{Y}-TS}
  are compact operators on X and Y respectively.
The index of a Fredholm operator is

  
    
      
        
          i
          n
          d
        
        
        T
        :=
        dim
        ⁡
        ker
        ⁡
        T
        −
        
          c
          o
          d
          i
          m
        
        
        
          r
          a
          n
        
        
        T
      
    
    {\displaystyle \mathrm {ind} \,T:=\dim \ker T-\mathrm {codim} \,\mathrm {ran} \,T}
  or in other words,

  
    
      
        
          i
          n
          d
        
        
        T
        :=
        dim
        ⁡
        ker
        ⁡
        T
        −
        
          d
          i
          m
        
        
        
          c
          o
          k
          e
          r
        
        
        T
        ;
      
    
    {\displaystyle \mathrm {ind} \,T:=\dim \ker T-\mathrm {dim} \,\mathrm {coker} \,T;}
  see dimension, kernel, codimension, range, and cokernel.