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

Bishop–Phelps theorem — Let $B\subseteq X$ be a bounded, closed, convex subset of a real Banach space $X.$ Then the set of all continuous linear functionals $f$ that achieve their supremum on $B$ (meaning that there exists some $b_0\in B$ such that $|f(b_0)|=\underset{b\in B}{\sup }|f(b)|$) $$\{f\in X^{*}:f\text{ attains its supremum on }B\}$$ is norm-dense in the continuous dual space $X^{*}$ of $X.$

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.

See also

• Banach–Alaoglu theorem – Theorem in functional analysis
• Dual norm – Measurement on a normed vector space
• Eberlein–Šmulian theorem – Relates three different kinds of weak compactness in a Banach space
• Goldstine theorem
• Mazur's lemma – On strongly convergent combinations of a weakly convergent sequence in a Banach space