Let $f$ be a non zero continuous linear functional on a Banach space $X$. i.e. $$f:X\rightarrow\mathbb{R}$$ is linear and bounded. Let $E$ be any non empty closed convex set of $X$ such that $$sup_{x\in E} |f(x)|$$ is attained. Then show that the supremum is attained at some extreme point of E.
Now if the assumption includes that the set $E$ is also compact then the supremum should be attaind (by the extreme value theorem) further by Krein- Milman theorem we can guarantee that it's attained at some extreme point.
But what about my question ? Is there any resource or book to find the proof ?
The statement as given is not true. Take $$ E = [0,1] \times \mathbb R $$ and $$ f(x,y)=x. $$ Then the supremum is attained on the line $\{1\}\times \mathbb R$. However $E$ has not extreme point at all.
You have to exclude by assumption that the minimum of $f$ is attained on a line, as lines do not tend to have extreme points.