I am trying to show that the closed unit ball in $C_0(R)$ has no extreme points.
This is what I got so far and I am stuck. Please help me.
Suppose that $f \in C_0(R)$ and is an extreme point of the closed unit ball.
Let $g(s) = f(s) + |f(s)| + 1$ and $h(s) = f(s) + |f(s)j| – 1$. Then $\|g\|_\infty \le 1$ and $\|h\|_\infty \le 1$ since $\|f\|_\infty \le 1$.
Is that the right way to start this problem?
Best Answer
I'm not sure what you are doing will lead anywhere.
Here's what you need to show:
If $f$ is in the unit ball of $C_0$, then there exists two functions $g$ and $h$ in the unit ball of $C_0$ such that $g\ne h$ and $f$ is a nontrivial convex combination of $g$ and $h$.
To show this, you could use the following
Hint: for $f$ in the unit ball of $C_0$, eventually $|f(x)|<1/2$. Using this, find two functions $g$ and $h$ in the unit ball of $C_0$ with $f={1\over2}(h+g)$.
Very informally, you can add a "bump" and subtract the same bump to $f$ over an interval on which the inequality first mentioned holds. $g$ and $h$ will have the forms $f+b$ and $f-b$, where $b$ is the "bump". Do this in such a way that the resulting functions are in the unit ball of $C_0$. Draw a picture here...
I hope that made sense...