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.^[1]

Statement

Its statement is as follows.

Bishop–Phelps theorem — Let ${\displaystyle B\subseteq X}$  be a bounded, closed, convex set of a real Banach space ${\displaystyle B\subseteq X.}$  Then the set

$${\displaystyle \left\{x^{\prime }\in X^{\prime }:x^{\prime }{\text{ attains its supremum on }}B\right\}}$$

 

is norm-dense in the continuous dual space ${\displaystyle X^{\prime }}$  of ${\displaystyle X.}$ 

Importantly, this theorem fails for complex Banach spaces.^[2]

See also

• Banach–Alaoglu theorem – Closed ball in the dual space is weak* compact
• Eberlein–Šmulian theorem – Relates three different kinds of weak compactness in a Banach space
• Mazur's lemma – On strongly convergent combinations of a weakly convergent sequence in a Banach space
• James' theorem
• Goldstine theorem

References

1. 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.
2. Lomonosov, Victor (2000). "A counterexample to the Bishop-Phelps theorem in complex spaces". Israel Journal of Mathematics. 115: 25–28. doi:10.1007/bf02810578. MR 1749671.