text: Q:
  
  Existence of extreme point in closed unit ball of normed space
  
  I just read the following quote from Conway:
  
  If $X$ is a Banach space, then $\text{ball }X^*$ is weak* compact, by Alaoglu's theorem. By the Krein-Milman Theorem, $\text{ball }X^*$ has many extreme points.
  
  Where $\text{ball }X^*$ is the closed unit call in $X^*$. However, I don't understand why we need $X$ to be Banach here, since Alaoglu's theorem holds for normed spaces, and all assumptions make in Krein-Milman hold either way too. I've been thinking about this for an hour at this point and I just don't see it.
  I would appreciate someone explaining why this does not hold for normed spaces. Thanks
  
  A:
  
  It does not matter. If $X$ is a normed space then the completion of $X$ say $V$ is a Banach space and $V^* = X^*. $ So w can always assume that $X$ is a Banach space.
  
  