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.

be a bounded, closed, convex subset of a real Banach space

Then the set of all continuous linear functionals

that achieve their supremum on

(meaning that there exists some

attains its supremum on

is norm-dense in the continuous dual space

Importantly, this theorem fails for complex Banach spaces.

is the closed unit ball then this theorem does hold for complex Banach spaces.

[1][2] This mathematical analysis–related article is a stub.