Linear isometry between $c_0$ and $c$

functional-analysis normed-spaces banach-spaces

Solution 1:

Hints:

  1. Prove that unit ball of $c$ have a lot of extreme points. In fact there are $\mathfrak{c}$ extreme points but this is not important for the solution.

  2. Prove that unit ball of $c_0$ have no extreme points.

  3. Prove that if $x$ is an extereme point of unit ball of some normed space $X$, and $i:X\to Y$ is a surjective isometry, then $i(x)$ is an extreme point of unit ball of $Y$.

  4. The rest is clear.

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