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. doi:10.1007/bf02810578.

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.CS1 maint: ref=harv (link)

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[1] Lomonosov, Victor (2000). "A counterexample to the Bishop-Phelps theorem in complex spaces". Israel J. Math. 115: 25–28. doi:10.1007/bf02810578.

Functional analysis (topics)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance

Bishop–Phelps theorem

See also

References