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 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 such that )

is norm-dense in the continuous dual space of

Importantly, this theorem fails for complex Banach spaces. However, for the special case where 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
• James' theorem – theorem in mathematics
• Goldstine theorem
• Mazur's lemma – On strongly convergent combinations of a weakly convergent sequence in a Banach space