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-1

1

$begingroup$

Consider the system
$$
T = { 1, cos(x), sin(x), dots, cos(nx), sin(nx), dots }
$$

I can show that they form a orthonormal system on $L^2( [ -pi, pi ] )$, but I don't know how to show how they form a complete system (for every $f in L^2( [ -pi, pi ]$, there is a unique representation in terms of $T$)

functional-analysis hilbert-spaces

share|cite|improve this question

asked yesterday

ketchupketchup

746

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Proof---web.mit.edu/jorloff/www/18.03-esg/notes/fourier-complete.pdf
  $endgroup$
  – herb steinberg
  yesterday
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This is the most fundamental result of the theory of Fourier series and every book on Fourier series has a proof. Some books (like the one by Edwards) has several proofs.
  $endgroup$
  – Kavi Rama Murthy
  yesterday
  

  
  
  
  

add a comment | 

-1

1

$begingroup$

Consider the system
$$
T = { 1, cos(x), sin(x), dots, cos(nx), sin(nx), dots }
$$

I can show that they form a orthonormal system on $L^2( [ -pi, pi ] )$, but I don't know how to show how they form a complete system (for every $f in L^2( [ -pi, pi ]$, there is a unique representation in terms of $T$)

functional-analysis hilbert-spaces

share|cite|improve this question

asked yesterday

ketchupketchup

746

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Proof---web.mit.edu/jorloff/www/18.03-esg/notes/fourier-complete.pdf
  $endgroup$
  – herb steinberg
  yesterday
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This is the most fundamental result of the theory of Fourier series and every book on Fourier series has a proof. Some books (like the one by Edwards) has several proofs.
  $endgroup$
  – Kavi Rama Murthy
  yesterday
  

  
  
  
  

add a comment | 

$begingroup$

Consider the system
$$
T = { 1, cos(x), sin(x), dots, cos(nx), sin(nx), dots }
$$

I can show that they form a orthonormal system on $L^2( [ -pi, pi ] )$, but I don't know how to show how they form a complete system (for every $f in L^2( [ -pi, pi ]$, there is a unique representation in terms of $T$)

functional-analysis hilbert-spaces

share|cite|improve this question

asked yesterday

ketchupketchup

746

$endgroup$

share|cite|improve this question

asked yesterday

ketchupketchup

746

share|cite|improve this question

asked yesterday

ketchupketchup

746

asked yesterday

ketchupketchup

746

asked yesterday

ketchupketchup

746