Title: "Bishop - Phelps Theorem and its relation with the RNP."

Abstract: Let $X$ be a complex Banach space and $C$ be a closed convex set in $X$. The set of support points of $C$ is the collection of points $z\in C$ for which there exists $T\in X^*$ such that $\sup_{x\in C}|T(x)| = |T(z)|$. The point $z$ is called a strongly exposed point, if for any sequence $z_n$ in $C$, if ${\rm Re\,} T(z_n)\rightarrow {\rm Re\,} T(z)$ then $z_n \rightarrow z$, where ${\rm Re\,}$ denotes the real part.

In 1991, Bishop asked if the Bishop-Phelps theorem does hold in the complex case. In 2000, Lomonosov constructed a closed bounded convex subset $C$ in a complex Banach space such that the set of support points of $C$ is empty. This means that the Bishop-Phelps theorem does not hold in the complex case. We will show that for Hardy space the Bishop-Phelps theorem does hold in the complex case.