Bishop–Phelps theorem explained

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

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.

Notes and References

1. Bishop. Errett. Errett Bishop. Phelps. R. R.. Robert R. Phelps. A proof that every Banach space is subreflexive. Bulletin of the American Mathematical Society. 67. 1961. 97–98. 123174. 10.1090/s0002-9904-1961-10514-4. free.
2. Lomonosov. Victor. Victor Lomonosov. A counterexample to the Bishop-Phelps theorem in complex spaces. Israel Journal of Mathematics. 2000. 115. 25–28. 10.1007/bf02810578. free. 1749671.