Let $X$ be a Banach space. By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball.

I am interested in its converse. More precisely,

Question: Let $X$ be a Banach space. If the closed unit ball of $X$ has at least one extreme point, must $X$ be a dual space?

I feel that the statement above is negative. However, I could not produce a counterexample.

In fact, the only Banach spaces which I know that are not dual spaces are $c_0$ and $C_0(\mathbb{R})$ (the latter set is the collection of all real-valued continuous function vanishing at infinity) because both sets have no extreme point.

closed unit ballof $X$ has at least on extreme point? $\endgroup$ – Martin Sleziak Dec 2 '18 at 7:11areinfinite compact $K$ for which $C(K)$ is a dual space: these are precisely the hyperstonean $K$, e.g., $\beta\mathbb{N}$. (On the other hand there are non-dual $C(K)$ for which the unit ball is the norm-closed convex hull of its extreme points, e.g. $\alpha\mathbb{N}$. These are precisely the totally disconected $K$.) $\endgroup$ – Dirk Werner Dec 2 '18 at 22:041more comments