Applied Analysis Seminar

**Monday February 25,
in K9509, SFU
**

3:30 - 4:20

Eldar Sadeghi

CECM, SFU

**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.