# 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.

How to Cite This Entry:
Bishop-Phelps theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bishop-Phelps_theorem&oldid=46075
This article was adapted from an original article by R. Phelps (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article