Bishop-Phelps theorem

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