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I was reading a book about duality and the examples of spaces that were not dual spaces were satisfying the following property: their unit ball had no extreme point.

I was, therefore, wondering about the following facts :

  • If the unit ball of a Banach space has an extreme point, is it necessarily the dual space of some other space?

  • Is there a Banach space such that the unit ball has extreme points but is not their closed convex hull?

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2 Answers 2

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Consider the space $C[0,1]$ of continuous real functions on the unit interval with the uniform norm. The unit Ball has extreme points, namely the constant functions with values $1$ and $-1$. But the closed convex hull of the extreme points under any separated linear topology is only a line segment and not the whole unit ball. In particular, $C[0,1]$ is not a dual space, for otherwise, the unit Ball would be the weak*-closed convex hull of the two extreme points and therefore a line segment.

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"Has an extreme point" is not enough. Not even "the unit ball is the closed convex hull of its extreme points" is enough. And it is not enough that "every closed bounded convex set is the closed convex hull of its extreme points". This "Krein-Milman property" can hold for a non-dual space.

Find lots more in the book Vector Measures by Diestel & Uhl.

For example, it is true that a separable dual space has the Krein-Milman property.

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