section 3 editing

1 files changed, 6 insertions, 3 deletions

diff --git a/Fourier Series.page b/Fourier Series.page
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--- a/Fourier Series.page
+++ b/Fourier Series.page
@@ -91,18 +91,21 @@ f & = & \Sigma e^{inx}\\
 $$  
   
 ##Definition of Inner Product of Functions  
-We begin proving this hypothesis by considering that any function on the right-hand side of our hypothesis is uniquely defined by the co-efficients of the terms $a_0$ through $a_n$ and $b_1$ through $b_n$. This can be taken to mean that every function is really a vector in an $2n+1$-dimensional 'Hilbert space'.(*perhaps someone can clarify this?*)  
+We begin proving this hypothesis by considering that any function on the right-hand side of our hypothesis is uniquely defined by the co-efficients of the terms $a_0$ through $a_n$ and $b_1$ through $b_n$. This can be taken to mean that every function is really a vector in a $2n+1$-dimensional 'Hilbert space'.(*perhaps someone can clarify this?*)  
   
 We now proceed to define certain operations on these functions in Hilbert space. One operation that will be particularly useful is that of the inner product of two functions:  
   
 ---> define inner product here
 
-This is the definition for a function of real numbers. For a function on complex numbers, the above definition must be altered as follows:   
+This is the inner product of 2 real-number functions. For a function on complex numbers, the above definition must be altered as follows:    
 --> altered function here    
   
 *Note: These are purely definitions, and we are now definining the inner product to ensure that inner product of f and f is a real number.*
 
-
+#Basis Vectors of the Hilbert Space  
+The basis vectors of this Hilbert space are taken as follows:  
+basis vectors, $f_n = \frac{1}{\sqrt{2\pi}}e^{inx}$  
+  
 
 #Why is Fourier series useful? </b>
 Applications will be covered on Monday July 5, 2010. See you all soon!  
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