Contracting Mapping

The better part of today's class was spent proving a theorem, which after a little bit of search I found is called the "Banach Fixed Point Theorem".

Banach Fixed Point Theorem:

Let (Y,m) me a complete metric space. Let T: Y -> Y be a continuous map such that for all x,y that are elements of Y, m(Tx,Ty) <= km(x,y) for some |k|<1.> x'.

From Mathworld:

Let be a contraction mapping from a closed subset of a Banach space into . Then there exists a unique such that .

My take on this:

It is easier for me to understand the theorem given to us in class rather than the theorem I found on Mathworld. (Maybe because I have no idea what "Banach space" is.) The way I see this is thatt T is a contracting transformation (which is the reason 0<1) style="font-weight: bold;">Summation of the Proof:

We proved this in 2 parts:

1. There is only 1 fixed point. This was proved by assuming there are 2 and showing they are equal. To do this we had to assume the sequence {(T^n)(y)} is cauchy.
   
2. Prove that the previous sequence is cauchy.