# 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.

Its statement is as follows.

Let B ⊂ E be a bounded, closed, convex set of a real Banach space E. Then the set
${\displaystyle \{e^{*}\in E^{*}\mid e^{*}{\text{ attains its supremum on }}B\}}$
is norm-dense in the dual ${\displaystyle E^{*}}$. Note, this theorem fails for complex Banach spaces [1]