$c$ and $c_0$ are not isometrically isomorphic I'm learning that the two Banach Spaces, $c$ and $c_0$ are not Isometrically Isomorphic.  
Where
$c$ = The set of all converging sequences. 
$c_0$ = The set of sequences converging to zero.  
Seems like the reason behind this is that the closed unit ball in two spaces have a different behavior. That is in $c$ they have got "extreme points" and in $c_0$ they do not.  
I'm a bit confused about this word "extreme point". First I thought that they were the boundary points but they cannot be because for sequences like $a=(1,0,0,0,0,...)$ in both $c_0$ and $c$, $||a||=\{\sup|a_n|: n\geq1\}=1$. So it is a boundary point of the closed balls in both spaces.  
Could some one help me out explaining about the extreme points in the given closed unit balls and how to determine whether they have got such points or not
 A: The idea is that a linear isometry $j: V\rightarrow W$ is injective and sends extreme points of any convex subset $K\subset V$ to extreme points of $j(K)\subset W$, 
in fact $j(K)$ is convex, for $j(u), j(v)\in j(K): tj(u)+(1-t)j(v)= j(tu+(1-t)v)\in j(K)$
Recall the definition of extreme points : $x$ is extreme in $C$ when $\exists\ a,b\in C \ x= ta+(1-t)b \implies a=b= x$
Then if $x\in K$ is extreme $$j(x)= tj(u)+(1-t)j(v)\implies j(x)=j(tu+(1-t)v) \implies x= tu+(1-t)v \implies u=v=x \implies j(u)=j(v)=j(x)$$ So $j(x)$ is extreme in $j(K)$ , and conversely as well if $j(y)$ is extreme then so is $y$ .
You can now apply this to $c_0$ and $c$, recall that every normed linear space is locally convex. In particular, every closed ball is convex.
The unit closed ball of $c_0$ has no extreme points, if $(x_n)$ is in this ball then $\exists N\in \mathbb{N}: \forall n\geq N\ |x_n|<1/5$ and we can construct $(y_n)$ and $(z_n)$ to be in this ball with $(x_n)= 1/2\left((y_n)+ (z_n)\right)$ by simply 'perturbing' just the $N^{th}$ term.
While the unit closed ball of $c$ has $(a_n)= (1,1,1,1,\dots)$ as an extreme point (why?) .
So having a different number of extreme points, what can you conclude? (be careful if you need your map to be surjective)
