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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.)
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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
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<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>
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Consider a real [[Banach space|Banach space]]  $  E $,
 +
its (closed convex) unit ball  $  B = \{ {y \in E } : {\| y \| \leq  1 } \} $,
 +
and its [[Adjoint space|adjoint space]] of continuous linear functionals  $  E  ^ {*} $(
 +
cf. [[Linear functional|Linear functional]]). If  $  x  ^ {*} \in E  ^ {*} $,
 +
its norm is defined as its supremum on the closed [[Convex set|convex set]]  $  B $,
 +
that is,  $  \| {x  ^ {*} } \| = \sup  \{ {x  ^ {*} ( y ) } : {\| y \| \leq  1 } \} $.
 +
The fundamental [[Hahn–Banach theorem|Hahn–Banach theorem]] implies that if  $  x \in E $
 +
and  $  \| x \| = 1 $,
 +
then there exists a continuous linear functional  $  x  ^ {*} \in E  ^ {*} $
 +
such that  $  x  ^ {*} ( x ) = 1 = \| {x  ^ {*} } \| $.
 +
Thus, these  "Hahn–Banach functionals" attain their suprema on  $  B $,
 +
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  $  E  ^ {*} $.
 +
(James' theorem [[#References|[a4]]] shows that if every element of  $  E  ^ {*} $
 +
attains its supremum on  $  B $,
 +
then  $  E $
 +
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  $  E  ^ {*} $
 +
which attain their supremum on an arbitrary non-empty closed convex bounded subset  $  C $
 +
of  $  E $(
 +
the support functionals of  $  C $).  
 +
In fact, if  $  C $
 +
is any non-empty closed convex subset of  $  E $,
 +
its support functionals are norm dense among those functionals which are bounded above on  $  C $;
 +
moreover, the points of  $  C $
 +
at which support functionals attain their supremum on  $  C $(
 +
the support points) are dense in the boundary of  $  C $.  
 +
(This contrasts with a geometric version of the Hahn–Banach theorem, which guarantees that every boundary point of a closed convex set  $  C $
 +
is a support point, provided  $  C $
 +
has non-empty interior.)
  
<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/b11058042.png" /></td> </tr></table>
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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  $  f $
 +
on  $  E $
 +
which are proper, in the sense that  $  - \infty < f \leq  \infty $
 +
and  $  f ( x ) < \infty $
 +
for at least one point  $  x $.
 +
The [[epigraph]]  $  { \mathop{\rm epi} } ( f ) = \{ {( x, r ) } : {x \in E, r \geq  f ( x ) } \} $
 +
of such a function is a non-empty closed convex subset of the product space  $  E \times \mathbf R $(
 +
$  \mathbf R $
 +
the real numbers) and the subgradients of  $  f $
 +
define support functionals of  $  { \mathop{\rm epi} } ( f ) $.  
 +
The set of all subgradients to  $  f $
 +
at  $  x $(
 +
where  $  f ( x ) $
 +
is finite) form the subdifferential
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058043.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058044.png" />. The Brøndsted–Rockafellar theorem [[#References|[a2]]] yields density, within the set of points where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058045.png" /> is finite, of those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058046.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110580/b11058047.png" /> is non-empty.
+
$$
 +
\partial  f ( x ) \equiv
 +
$$
 +
 
 +
$$
 +
\equiv
 +
\left \{ {x  ^ {*} \in E  ^ {*} } : {x  ^ {*} ( y - x ) \leq  f ( y ) - f ( x )  \textrm{ for  all  }  y \in E } \right \}
 +
$$
 +
 
 +
of $  f $
 +
at $  x $.  
 +
The Brøndsted–Rockafellar theorem [[#References|[a2]]] yields density, within the set of points where $  f $
 +
is finite, of those $  x $
 +
for which $  \partial  f ( x ) $
 +
is non-empty.
  
 
See also [[#References|[a3]]] for the Bishop–Phelps and James theorems, [[#References|[a5]]] for the Bishop–Phelps and Brøndsted–Rockafellar theorems.
 
See also [[#References|[a3]]] for the Bishop–Phelps and James theorems, [[#References|[a5]]] for the Bishop–Phelps and Brøndsted–Rockafellar theorems.
Line 18: Line 80:
 
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R.C. James,  "Reflexivity and the supremum of linear functionals"  ''Israel J. Math.'' , '''13'''  (1972)  pp. 289–300</TD></TR>
 
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R.C. James,  "Reflexivity and the supremum of linear functionals"  ''Israel J. Math.'' , '''13'''  (1972)  pp. 289–300</TD></TR>
 
<TR><TD valign="top">[a5]</TD> <TD valign="top">  R.R. Phelps,  "Convex functions, monotone operators and differentiability" , ''Lecture Notes in Mathematics'' , '''1364''' , Springer  (1993)  (Edition: Second)</TD></TR>
 
<TR><TD valign="top">[a5]</TD> <TD valign="top">  R.R. Phelps,  "Convex functions, monotone operators and differentiability" , ''Lecture Notes in Mathematics'' , '''1364''' , Springer  (1993)  (Edition: Second)</TD></TR>
<TR><TD valign="top">[b1]</TD> <TD valign="top">  Andrzej Granas, James Dugundji, "Fixed Point Theory", Springer Monographs in Mathematics, Springer (2003) ISBN 0-387-00173-5 {{ZBL|1025.47002}}</TD></TR>
+
<TR><TD valign="top">[b1]</TD> <TD valign="top">  Andrzej Granas, James Dugundji, "Fixed Point Theory", Springer Monographs in Mathematics, Springer (2003) {{ISBN|0-387-00173-5}} {{ZBL|1025.47002}}</TD></TR>
 
</table>
 
</table>

Latest revision as of 19:35, 6 December 2023


Consider a real Banach space $ E $, its (closed convex) unit ball $ B = \{ {y \in E } : {\| y \| \leq 1 } \} $, and its adjoint space of continuous linear functionals $ E ^ {*} $( cf. Linear functional). If $ x ^ {*} \in E ^ {*} $, its norm is defined as its supremum on the closed convex set $ B $, that is, $ \| {x ^ {*} } \| = \sup \{ {x ^ {*} ( y ) } : {\| y \| \leq 1 } \} $. The fundamental Hahn–Banach theorem implies that if $ x \in E $ and $ \| x \| = 1 $, then there exists a continuous linear functional $ x ^ {*} \in E ^ {*} $ such that $ x ^ {*} ( x ) = 1 = \| {x ^ {*} } \| $. Thus, these "Hahn–Banach functionals" attain their suprema on $ B $, 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 $ E ^ {*} $. (James' theorem [a4] shows that if every element of $ E ^ {*} $ attains its supremum on $ B $, then $ E $ is necessarily reflexive, cf. Reflexive space.) A more general Bishop–Phelps theorem yields the same norm density conclusion for the set of functionals in $ E ^ {*} $ which attain their supremum on an arbitrary non-empty closed convex bounded subset $ C $ of $ E $( the support functionals of $ C $). In fact, if $ C $ is any non-empty closed convex subset of $ E $, its support functionals are norm dense among those functionals which are bounded above on $ C $; moreover, the points of $ C $ at which support functionals attain their supremum on $ C $( the support points) are dense in the boundary of $ C $. (This contrasts with a geometric version of the Hahn–Banach theorem, which guarantees that every boundary point of a closed convex set $ C $ is a support point, provided $ C $ 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 $ f $ on $ E $ which are proper, in the sense that $ - \infty < f \leq \infty $ and $ f ( x ) < \infty $ for at least one point $ x $. The epigraph $ { \mathop{\rm epi} } ( f ) = \{ {( x, r ) } : {x \in E, r \geq f ( x ) } \} $ of such a function is a non-empty closed convex subset of the product space $ E \times \mathbf R $( $ \mathbf R $ the real numbers) and the subgradients of $ f $ define support functionals of $ { \mathop{\rm epi} } ( f ) $. The set of all subgradients to $ f $ at $ x $( where $ f ( x ) $ is finite) form the subdifferential

$$ \partial f ( x ) \equiv $$

$$ \equiv \left \{ {x ^ {*} \in E ^ {*} } : {x ^ {*} ( y - x ) \leq f ( y ) - f ( x ) \textrm{ for all } y \in E } \right \} $$

of $ f $ at $ x $. The Brøndsted–Rockafellar theorem [a2] yields density, within the set of points where $ f $ is finite, of those $ x $ for which $ \partial f ( x ) $ 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