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Consider a real [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105801.png" />, its (closed convex) unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105802.png" />, and its [[Adjoint space|adjoint space]] of continuous linear functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105803.png" /> (cf. [[Linear functional|Linear functional]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105804.png" />, its norm is defined as its supremum on the closed [[Convex set|convex set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105805.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105806.png" />. The fundamental [[Hahn–Banach theorem|Hahn–Banach theorem]] implies that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105807.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105808.png" />, then there exists a continuous linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105809.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058010.png" />. Thus, these  "Hahn–Banach functionals"  attain their suprema on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058011.png" />, and by taking all positive scalar multiples of such functions, there are clearly  "many"  of them. The Bishop–Phelps theorem [[#References|[a1]]] asserts that such norm-attaining functionals are actually norm dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058012.png" />. (James' theorem [[#References|[a4]]] shows that if every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058013.png" /> attains its supremum on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058015.png" /> is necessarily reflexive, cf. [[Reflexive space|Reflexive space]].) A more general Bishop–Phelps theorem yields the same norm density conclusion for the set of functionals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058016.png" /> which attain their supremum on an arbitrary non-empty closed convex bounded subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058018.png" /> (the support functionals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058019.png" />). In fact, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058020.png" /> is any non-empty closed convex subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058021.png" />, its support functionals are norm dense among those functionals which are bounded above on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058022.png" />; moreover, the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058023.png" /> at which support functionals attain their supremum on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058024.png" /> (the support points) are dense in the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058025.png" />. (This contrasts with a geometric version of the Hahn–Banach theorem, which guarantees that every boundary point of a closed convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058026.png" /> is a support point, provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058027.png" /> has non-empty interior.)
 
Consider a real [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105801.png" />, its (closed convex) unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105802.png" />, and its [[Adjoint space|adjoint space]] of continuous linear functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105803.png" /> (cf. [[Linear functional|Linear functional]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105804.png" />, its norm is defined as its supremum on the closed [[Convex set|convex set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105805.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105806.png" />. The fundamental [[Hahn–Banach theorem|Hahn–Banach theorem]] implies that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105807.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105808.png" />, then there exists a continuous linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b1105809.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058010.png" />. Thus, these  "Hahn–Banach functionals"  attain their suprema on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058011.png" />, and by taking all positive scalar multiples of such functions, there are clearly  "many"  of them. The Bishop–Phelps theorem [[#References|[a1]]] asserts that such norm-attaining functionals are actually norm dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058012.png" />. (James' theorem [[#References|[a4]]] shows that if every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058013.png" /> attains its supremum on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058015.png" /> is necessarily reflexive, cf. [[Reflexive space|Reflexive space]].) A more general Bishop–Phelps theorem yields the same norm density conclusion for the set of functionals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058016.png" /> which attain their supremum on an arbitrary non-empty closed convex bounded subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058018.png" /> (the support functionals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058019.png" />). In fact, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058020.png" /> is any non-empty closed convex subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058021.png" />, its support functionals are norm dense among those functionals which are bounded above on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058022.png" />; moreover, the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058023.png" /> at which support functionals attain their supremum on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058024.png" /> (the support points) are dense in the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058025.png" />. (This contrasts with a geometric version of the Hahn–Banach theorem, which guarantees that every boundary point of a closed convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058026.png" /> is a support point, provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058027.png" /> has non-empty interior.)
  
This last result leads to the [[Brøndsted–Rockafellar theorem|Brøndsted–Rockafellar theorem]] [[#References|[a2]]], fundamental in [[Convex analysis|convex analysis]], about extended-real-valued lower semi-continuous convex functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058028.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058029.png" /> which are proper, in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058031.png" /> for at least one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058032.png" />. The epigraph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058033.png" /> of such a function is a non-empty closed convex subset of the product space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058034.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058035.png" /> the real numbers) and the subgradients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058036.png" /> define support functionals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058037.png" />. The set of all subgradients to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058038.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058039.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058040.png" /> is finite) form the subdifferential
+
This last result leads to the [[Brøndsted–Rockafellar theorem|Brøndsted–Rockafellar theorem]] [[#References|[a2]]], fundamental in [[Convex analysis|convex analysis]], about extended-real-valued lower semi-continuous convex functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058028.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058029.png" /> which are proper, in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058031.png" /> for at least one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058032.png" />. The [[epigraph]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058033.png" /> of such a function is a non-empty closed convex subset of the product space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058034.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058035.png" /> the real numbers) and the subgradients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058036.png" /> define support functionals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058037.png" />. The set of all subgradients to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058038.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058039.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058040.png" /> is finite) form the subdifferential
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058041.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058041.png" /></td> </tr></table>

Revision as of 17:03, 7 May 2017

Consider a real Banach space , its (closed convex) unit ball , and its adjoint space of continuous linear functionals (cf. Linear functional). If , its norm is defined as its supremum on the closed convex set , that is, . The fundamental Hahn–Banach theorem implies that if and , then there exists a continuous linear functional such that . Thus, these "Hahn–Banach functionals" attain their suprema on , and by taking all positive scalar multiples of such functions, there are clearly "many" of them. The Bishop–Phelps theorem [a1] asserts that such norm-attaining functionals are actually norm dense in . (James' theorem [a4] shows that if every element of attains its supremum on , then is necessarily reflexive, cf. Reflexive space.) A more general Bishop–Phelps theorem yields the same norm density conclusion for the set of functionals in which attain their supremum on an arbitrary non-empty closed convex bounded subset of (the support functionals of ). In fact, if is any non-empty closed convex subset of , its support functionals are norm dense among those functionals which are bounded above on ; moreover, the points of at which support functionals attain their supremum on (the support points) are dense in the boundary of . (This contrasts with a geometric version of the Hahn–Banach theorem, which guarantees that every boundary point of a closed convex set is a support point, provided has non-empty interior.)

This last result leads to the Brøndsted–Rockafellar theorem [a2], fundamental in convex analysis, about extended-real-valued lower semi-continuous convex functions on which are proper, in the sense that and for at least one point . The epigraph of such a function is a non-empty closed convex subset of the product space ( the real numbers) and the subgradients of define support functionals of . The set of all subgradients to at (where is finite) form the subdifferential

of at . The Brøndsted–Rockafellar theorem [a2] yields density, within the set of points where is finite, of those for which is non-empty.

See also [a3] for the Bishop–Phelps and James theorems, [a5] for the Bishop–Phelps and Brøndsted–Rockafellar theorems.

References

[a1] E. Bishop, R.R. Phelps, "The support functionals of a convex set" P. Klee (ed.) , Convexity , Proc. Symp. Pure Math. , 7 , Amer. Math. Soc. (1963) pp. 27–35 Zbl 0149.08601
[a2] A. Brøndsted, R.T. Rockafellar, "On the subdifferentiability of convex functions" Proc. Amer. Math. Soc. , 16 (1965) pp. 605–611
[a3] J. Diestel, "Geometry of Banach spaces: Selected topics" , Lecture Notes in Mathematics , 485 , Springer (1975)
[a4] R.C. James, "Reflexivity and the supremum of linear functionals" Israel J. Math. , 13 (1972) pp. 289–300
[a5] R.R. Phelps, "Convex functions, monotone operators and differentiability" , Lecture Notes in Mathematics , 1364 , Springer (1993) (Edition: Second)
[b1] Andrzej Granas, James Dugundji, "Fixed Point Theory", Springer Monographs in Mathematics, Springer (2003) ISBN 0-387-00173-5 Zbl 1025.47002
How to Cite This Entry:
Bishop-Phelps theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bishop-Phelps_theorem&oldid=41307
This article was adapted from an original article by R. Phelps (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article