started section 1

1 files changed, 11 insertions, 3 deletions

diff --git a/ClassJune28.page b/ClassJune28.page
index b3bd725..e3e4874 100644
--- a/ClassJune28.page
+++ b/ClassJune28.page
@@ -4,14 +4,22 @@
 
 #<b>Why the Fourier decomposition is possible?</b>
 
-We first begin with a few basic identities on the size of sets. Then, we will show that the set of possible functions representing sets is not larger than the set of available functions. This at best indicates that the Fourier series is not altogether impossible.  
-  
+A Fourier series is a function of the form $\sum_{n=-\infty}^\infty a_n e^{inx}$ or $\sum_{n=-\infty}^\infty b_n \cos(n x) + c_n \sin (n x)$, depending on one's taste for the imaginary. The rumor on the street is that any periodic function (well, any nice one) can be expressed as a Fourier series: you hand me a function $f:[0,2\pi]\rightarrow \mathbb{C}$ and I hand you a list of real numbers $b_0,c_0,b_1,c_1,b_2,c_2,b_3,c_3,\dots$ such that
+
+$f(x) = \sum_{n=-\infty}^\infty b_n \cos(n x) + c_n \sin (n x).$
+
+If this exchange is always possible, there must be at least as many different lists of real numbers as there are nice periodic functions. So how many are there of each?
+
+Infinity, of course. So to perform a meaningful check that the set of nice periodic functions is the same size as the set of lists of real numbers, we need a more refined notion of a size of a set than just giving its number of elements, or saying that it is infinite. We say that two sets have the same cardinality (our new word for size) if we can pair off their elements, giving a perfect matching between the sets.
+
+
 ## To show that $(0,1) \sim \mathbb R$
 *--> could someone fill this out?  *
 
 ## Cantor's proof for $\mathbb R > \mathbb N$
-*--> don't have the notes for this  *
+Suppose that we had a bijection between $\mathbb N$ and $\mathbb R$. Since $\mathbb R \sim \mathbb N$ In other words, we have a list of all the real numbers
 
+1-->0.
 ## Proof that no. of available functions is greater than number of functions required to define the periodic function
 Consider any arbitrary periodic function in the interval $[-\pi,\pi]$. This can be represented as a series of values at various points in the interval. For example,  
 $\qquad f(0) = ... , f(0.1) = ..., f(0.2) = ...$ and so on. At each point, we can assign any real number (i.e. $\in \mathbb R$). So, the number of possible periodic functions in an interval is of the order of $\mathbb R^{\mathbb R}$.