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. 



If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?
Every separable Banach space $X$ can be equivalently renormed so that every point in the unit sphere is an extreme point: Take an injective bounded linear operator $T$ from $X$ into $\ell\_2$ and use $|x| := \|x\|\_X + \|Tx\|\_2$. Of course, there are many separable Banach spaces that are not isomorphic to a separable conjugate space, including (as Dirk pointed out) those that fail the Radon Nikodym property.


