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 [math]\displaystyle{ B \subseteq X }[/math] be a bounded, closed, convex subset of a real Banach space [math]\displaystyle{ X. }[/math] Then the set of all continuous linear functionals [math]\displaystyle{ f }[/math] that achieve their supremum on [math]\displaystyle{ B }[/math] (meaning that there exists some [math]\displaystyle{ b_0 \in B }[/math] such that [math]\displaystyle{ |f(b_0)| = \sup_{b \in B} |f(b)| }[/math]) [math]\displaystyle{ \left\{f \in X^* : f \text{ attains its supremum on } B\right\} }[/math] is norm-dense in the continuous dual space [math]\displaystyle{ X^* }[/math] of [math]\displaystyle{ X. }[/math]

Importantly, this theorem fails for complex Banach spaces. However, for the special case where [math]\displaystyle{ B }[/math] 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