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
Bishop–Phelps theorem — Let [math]\displaystyle{ B \subseteq X }[/math] be a bounded, closed, convex subset of a real Banach space [math]\displaystyle{ X. }[/math] Then the set of all continuous linear functionals [math]\displaystyle{ f }[/math] that achieve their supremum on [math]\displaystyle{ B }[/math] (meaning that there exists some [math]\displaystyle{ b_0 \in B }[/math] such that [math]\displaystyle{ |f(b_0)| = \sup_{b \in B} |f(b)| }[/math]) [math]\displaystyle{ \left\{f \in X^* : f \text{ attains its supremum on } B\right\} }[/math] is norm-dense in the continuous dual space [math]\displaystyle{ X^* }[/math] of [math]\displaystyle{ X. }[/math]
Importantly, this theorem fails for complex Banach spaces.[2] However, for the special case where [math]\displaystyle{ B }[/math] is the closed unit ball then this theorem does hold for complex Banach spaces.[1][2]
See also
- Banach–Alaoglu theorem – Theorem in functional analysis
- Dual norm – Measurement on a normed vector space
- Eberlein–Šmulian theorem – Relates three different kinds of weak compactness in a Banach space
- Goldstine theorem
- Mazur's lemma – On strongly convergent combinations of a weakly convergent sequence in a Banach space
References
- ↑ 1.0 1.1 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.
- ↑ 2.0 2.1 Lomonosov, Victor (2000). "A counterexample to the Bishop-Phelps theorem in complex spaces". Israel Journal of Mathematics 115: 25–28. doi:10.1007/bf02810578.
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