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 be a bounded, closed, convex set of a real Banach space Then the set
is norm-dense in the continuous dual space of
Importantly, this theorem fails for complex Banach spaces.[2]
See also[]
- Banach–Alaoglu theorem – The closed unit ball in the dual of a normed vector space is compact in the weak* topology
- 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[]
- ^ 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.
- ^ 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.
Categories:
- Theorems in functional analysis
- Mathematical analysis stubs