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.

Its statement is as follows.

Let B ⊂ E be a bounded, closed, convex set of a real Banach space E. Then the set

\{e^{*}\in E^{*}\mid e^{*}{\text{ attains its supremum on }}B\}

is norm-dense in the dual

E^{*}

. Note, this theorem fails for complex Banach spaces ^[1]

References

↑ Lomonosov, Victor (2000). "A counterexample to the Bishop-Phelps theorem in complex spaces". Israel J. Math. 115: 25–28.

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 123174.

Functional analysis

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Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

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Bishop–Phelps theorem

See also

References