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 Banach space E. Then the set

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

is norm-dense in the dual $E^{*}$.

See also

• Banach–Alaoglu theorem
• Eberlein–Šmulian theorem
• Mazur's lemma
• James' theorem
• Goldstine theorem

References

• 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. {{cite journal}}: Invalid |ref=harv (help)