View source for Bishop–Phelps theorem

In mathematics, the '''Bishop–Phelps theorem''' is a theorem about the topological properties of [[Banach space]]s named after [[Biography:Errett Bishop|Errett Bishop]] and [[Biography:Robert Phelps|Robert Phelps]], who published its proof in 1961.<ref name="BishopPhelps1961">{{cite journal|last1=Bishop|first1=Errett|last2=Phelps|first2=R. R.|author-link2=Robert R. Phelps|title=A proof that every Banach space is subreflexive|journal=Bulletin of the American Mathematical Society|volume=67|year=1961|pages=97–98|mr=123174|doi=10.1090/s0002-9904-1961-10514-4|doi-access=free}}</ref>

==Statement==

{{math theorem|name=Bishop–Phelps theorem|note=|style=|math_statement=
Let <math>B \subseteq X</math> be a bounded, closed, convex subset of a real [[Banach space]] <math>X.</math> Then the set of all continuous linear functionals <math>f</math> that achieve their supremum on <math>B</math> (meaning that there exists some <math>b_0 \in B</math> such that <math>|f(b_0)| = \sup_{b \in B} |f(b)|</math>) 
<math display=block>\left\{f \in X^* : f \text{ attains its supremum on } B\right\}</math> 
is [[Dual norm|norm]]-dense in the continuous dual space <math>X^*</math> of <math>X.</math> 
}}

Importantly, this theorem fails for complex Banach spaces.<ref name="Lomonosov2000">{{cite journal|last1=Lomonosov|first1=Victor|title=A counterexample to the Bishop-Phelps theorem in complex spaces|journal=[[Israel Journal of Mathematics]]|date=2000|volume=115|pages=25–28|doi=10.1007/bf02810578|doi-access=free|mr=1749671}}</ref> 
However, for the special case where <math>B</math> is the closed unit ball then this theorem does hold for complex Banach spaces.<ref name="BishopPhelps1961" /><ref name="Lomonosov2000" />

==See also==

* {{annotated link|Banach–Alaoglu theorem}}
* {{annotated link|Dual norm}}
* {{annotated link|Eberlein–Šmulian theorem}}
* {{annotated link|Goldstine theorem}}
* {{annotated link|Mazur's lemma}}

==References==

{{reflist}}

{{Functional analysis}}

{{DEFAULTSORT:Bishop-Phelps theorem}}

[[Category:Banach spaces]]
[[Category:Theorems in functional analysis]]

{{Sourceattribution|Bishop–Phelps theorem}}