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.^[1]

Statement

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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 $|f(b_{0})|=\sup _{b\in B}|f(b)|$ ) $\left\{f\in X^{*}:f{\text{ attains its supremum on }}B\right\}$ is norm-dense in the continuous dual space $X^{*}$ of $X.$

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

References

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^ ^a ^b Bishop, Errett; Phelps, R. R. (1961). "A proof that every Banach space is subreflexive". Bulletin of the American Mathematical Society. 67: 97–98. doi:10.1090/s0002-9904-1961-10514-4. MR 0123174.
^ ^a ^b Lomonosov, Victor (2000). "A counterexample to the Bishop-Phelps theorem in complex spaces". Israel Journal of Mathematics. 115: 25–28. doi:10.1007/bf02810578. MR 1749671.

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