I am trying to prove that $\mathcal{c}$ and $\mathcal{c_0}$ are isomorphic but not isometrically isomorphic.
I've read on this forum that isometric isomorphism preserves extreme points, but I don't see why. I know that I should use the fact that isometries preserve all metric properties and being an extreme point is a metric property but I can't come up with any proof.
If $K$ is a convex set and $a \in K$ is its extreme point, then whenever $a = \frac{x+y}{2}, \ x,y \in K, $ we have that $x=y=a$.
What should I do now?
Could you help me with that?
Thanks.