I'm studying for an exam and was looking for some good questions in some outside textbooks. I found this problem in Ciarlet's functional textbook:
Let $X$ be a normed vector space and let $f : X \to \mathbb{R}$ be a convex and continuous function. Show that there exists $l \in X'$ and $c \in \mathbb{R}$ such that $f(x) > l(x) + c$ for all $x \in X$
My thoughts: Drawing a quick sketch, the intuition for why this is true is clear. My idea was to consider $A := \text{im}(f)$, and define $M := \{tf(x)+(1-t)f(y) : x,y \in X \text{ and } t \in [0,1]\}$ and try to apply Hahn-Banach Separation theorem, but there are two problems with this: (1) we cannot guarantee the openness or closedness of $M$ and (2) using HB here would give us a separating hyperplane defined by a linear functional whose domain is in $\mathbb{R}$. So this leaves me with trying to be clever by choosing some particular sets in $X$ to separate (e.g. I tried looking at level sets of $f$), but I haven't been able to make much progress.
Any help would be much appreciated! Thanks!