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.

Bishop–Phelps theorem

See also

References