section 3 editing

1 files changed, 12 insertions, 1 deletions

diff --git a/Fourier Series.page b/Fourier Series.page
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+++ b/Fourier Series.page
@@ -90,6 +90,17 @@ f & = & \Sigma e^{inx}\\
 $$  
   
   
-We begin proving this hypothesis by  
+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 n-dimensional Hilbert space.  
+  
+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 in Hilbert space:  
+  
+---> 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:   
+  
+  
+*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.*
+
+
 #Why is Fourier series useful? </b>
 Applications will be covered on Monday July 5, 2010. See you all soon!  
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