Bishop–Phelps theorem

In mathematics, the Bishop–Phelps theorem is a theorem about the topological properties of Banach spaces named after Errett Bishop and Robert Phelps, who published its proof in 1961.

Statement
$$

Importantly, this theorem fails for complex Banach spaces. However, for the special case where $$B$$ is the closed unit ball then this theorem does hold for complex Banach spaces.