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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
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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 |
[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) |
Bishop-Phelps theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bishop-Phelps_theorem&oldid=17533