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.^[1]

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 ${\displaystyle X.}$

Importantly, this theorem fails for complex Banach spaces.^[2] However, for the special case where ${\displaystyle B}$ is the closed unit ball then this theorem does hold for complex Banach spaces.^[1][2]

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• Goldstine theorem
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