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Difference between revisions of "Bishop-Phelps theorem"

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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Brøndsted,  R.T. Rockafellar,  "On the subdifferentiability of convex functions"  ''Proc. Amer. Math. Soc.'' , '''16'''  (1965)  pp. 605–611</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Diestel,  "Geometry of Banach spaces: Selected topics" , ''Lecture Notes in Mathematics'' , '''485''' , Springer  (1975)</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></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  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}}</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Brøndsted,  R.T. Rockafellar,  "On the subdifferentiability of convex functions"  ''Proc. Amer. Math. Soc.'' , '''16'''  (1965)  pp. 605–611</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Diestel,  "Geometry of Banach spaces: Selected topics" , ''Lecture Notes in Mathematics'' , '''485''' , Springer  (1975)</TD></TR>
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<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">[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>
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</table>

Revision as of 18:01, 10 January 2015

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=36197
This article was adapted from an original article by R. Phelps (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article