Bishop–Phelps theorem

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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

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Bishop–Phelps theorem — Let   be a bounded, closed, convex subset of a real Banach space   Then the set of all continuous linear functionals   that achieve their supremum on   (meaning that there exists some   such that  )   is norm-dense in the continuous dual space   of  

Importantly, this theorem fails for complex Banach spaces.[2] However, for the special case where   is the closed unit ball then this theorem does hold for complex Banach spaces.[1][2]

See also

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References

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  1. ^ a b 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. ^ a b 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.