The Bishop-Phelps Theorem asserts that the set of functionals which attain the maximum value on a closed bounded convex subset of a real Banach space is norm dense in the dual space. We show that this statement cannot be extended to general complex Banach spaces by constructing a closed bounded convex set with no support points. We show also that if the Bishop-Phelps Theorem is correct for a uniform dual algebra R of operators in a Hilbert space,
then the algebra R is selfadjoint.