Question

(Lindenstrauss, Phelps) Show that if \(X\) is an infinite-dimensional reflexive Banach space, then \(\operatorname{Ext}\left(B_{X}\right)\) is uncountable.

Short answer

Since the extreme points must form an uncountable set to maintain the weakly compact structure in infinite dimensions, \( \text{Ext}(B_X) \) is uncountable.

Step-by-step solution

- Understand Key Terms

Familiarize with key terms: 1. A Banach space is a complete normed vector space. 2. A space is reflexive if the natural map from the space into its double dual is an isometry. 3. \(\text{Ext}(B_X)\) denotes the set of extreme points of the unit ball of X.

- Establishing the Extreme Points

The Krein-Milman theorem states that the closed convex hull of the set of extreme points of a compact convex set in a topological vector space is the set itself. In an infinite-dimensional reflexive Banach space, the unit ball \(B_X\) is not compact, but its dual ball in \(X^*\) is compact in the weak* topology.

- Analyze Fundamental Properties

Realize that in a reflexive Banach space, both the unit ball and its dual are related via the weak* topology, and consider the consequences on the geometric structure of the unit ball.

- Counting Argument

An important aspect is that if \( \text{Ext}(B_X) \) were countable, \( B_{X^*} \) (the unit ball of the dual space) would not remain weakly compact given the Banach-Alaoglu theorem since a countable weakly compact set in infinite-dimensional normed spaces would contradict Kakutani's theorem.

- Conclusion

Because a countable set cannot make up the extreme points of a weakly compact set in infinite dimensions due to the above arguments, \( \text{Ext}(B_X) \) must be uncountable, thus affirming the required condition.

Key concepts

Extreme Points

To start, an extreme point of a convex set is a point in the set that cannot be written as a convex combination of two other distinct points in the set. For example, the corners of a polygon in the plane are its extreme points. In the context of Banach spaces, the extreme points of the unit ball play a crucial role. The set of extreme points of the unit ball of a Banach space, denoted as \text{Ext}(B_X), helps us understand more about the structure of the space. The problem asks us to show that if \(X\) is an infinite-dimensional reflexive Banach space, then \text{Ext}(B_X) is uncountable.

Krein-Milman Theorem

The Krein-Milman theorem is a fundamental result in functional analysis. It states that a compact convex subset of a topological vector space is the closed convex hull of its extreme points. That means every compact convex set is completely determined by its extreme points. For an infinite-dimensional reflexive Banach space, although the unit ball \(B_X\) itself is not compact, its dual ball, \(B_{X^*}\), is compact in the weak* topology. This insight is essential in understanding how extreme points and the structure of the unit ball are related in these spaces.

Weak* Topology

The weak* topology (or weak-star topology) is the topology on the dual space of a Banach space, denoted \(X^*\), where a net \((f_u)\) converges to \(f\) if and only if \(f_u(x) \to f(x)\) for every \(x \text{ in } X\). This topology is coarser (weaker) than the norm topology of \(X^*\), meaning that fewer sets are open in the weak* topology compared to the norm topology. In our problem, we consider the unit ball of \(X^*\), \(B_{X^*}\), which is compact in the weak* topology due to the Banach-Alaoglu theorem.

Banach-Alaoglu Theorem

The Banach-Alaoglu theorem guarantees that the closed unit ball in the dual of a normed space, \(B_{X^*}\), is compact in the weak* topology. This compactness does not imply the unit ball is compact in the norm topology but provides significant geometric insights. This property becomes particularly useful in infinite-dimensional reflexive Banach spaces, as it links the unit ball of the space to its geometric properties via extreme points, aiding in the counting argument needed to demonstrate uncountability.

Kakutani's Theorem

Kakutani's theorem states that in certain conditions, weakly compact sets coincide with norm-compact sets in dual spaces. More relevant to our context, for infinite-dimensional Banach spaces, a weak compactness helps in deducing important properties about the bounded sets. Using Kakutani's theorem, we realize that a countable set cannot form the extreme points of a weakly compact set in an infinite-dimensional space. Therefore, the assumption that \( \text{Ext}(B_X) \) is countable leads to a contradiction, proving that \(\text{Ext}(B_X)\) must be uncountable.