Bishop–Phelps theorem

Bishop–Phelps theorem — Let ${\displaystyle B\subseteq X}$  be a bounded, closed, convex subset of a real Banach space ${\displaystyle X.}$  Then the set of all continuous linear functionals ${\displaystyle f}$  that achieve their supremum on ${\displaystyle B}$  (meaning that there exists some ${\displaystyle b_{0}\in B}$  such that ${\displaystyle |f(b_{0})|=\sup _{b\in B}|f(b)|}$ ) $${\displaystyle \left\{f\in X^{*}:f{\text{ attains its supremum on }}B\right\}}$$  is norm-dense in the continuous dual space ${\displaystyle X^{*}}$  of ${\displaystyle X.}$