## Joint Math-Stat Colloquium: Pradipta Bandyopadhyay

#### Distingushed Speaker from the Indian Statistical Institute

Friday, May 17, 2024 · 11 AM - 12 PM

**Title**: On The Krein-Milman Property in Banach Spaces

**Abstract**: A Banach space X is said to have the Krein-Milman Property (KMP) if every nonempty closed bounded convex set K in X is the closed convex hull of its extreme points.

This property gets its name from the classical Krein-Milman Theorem. It follows that finite dimensional or reflexive spaces have the KMP.

We give a few examples of Banach spaces without the KMP.

We recall the proof of the Krein-Milman Theorem and show that the proof can be adapted to obtain a sufficient condition for the KMP.

This sufficient condition is related to the Radon-Nikody'm Property (RNP), which says roughly that the Radon-Nikody'm Theorem holds for vector measures taking values in X.

We will have the Departmental Coffee and Tea from 10 to 10:45 in M&P 422.