Question

([Mon2]) Let \(X\) be a Banach space. The drop defined by \(x \in X \backslash B_{X}\) is the set \(D\left(x, B_{X}\right)=\operatorname{conv}\left(\\{x\\} \cup B_{X}\right) .\) The Banach space \(X\) is said to have the drop property if, given any closed set \(S \subset X\) such that \(S \cap B_{X}=\emptyset\), there exists \(s \in S\) such that \(D\left(s, B_{X}\right) \cap S=\\{x\\}\) (i) Let \(X\) be a Banach space. Show that, given a closed set \(A \subset X\) such that \(\operatorname{dist}\left(A, B_{X}\right)>0\), there exists \(a \in A\) such that \(D\left(a, B_{X}\right) \cap A=\\{a\\}\) ([Dan]). (ii) Prove that \(X\) has the drop property if and only if \(X\) is reflexive and has the Kadec property (that is, norm and weak convergent sequences in \(S_{X}\) are the same).

Short answer

(i) There exists a in A such that D(a, B_X) ∩ A = {a}. (ii) X has the drop property iff X is reflexive and has the Kadec property.

Step-by-step solution

Understand the definition of the drop

For any Banach space X and an element x in the space but not in the closed unit ball, the drop defined by x and B_X is the convex hull of x union B_X, denoted as D(x, B_X) = conv({x} ∪ B_X).

Set up the condition for part (i)

Consider a closed set A in X such that the distance between A and the unit ball B_X is greater than zero, which implies that elements of A are outside B_X.

Prove the existence of a specific element in A for part (i)

By the given condition, dist(A, B_X) > 0, meaning there is a positive distance. Hence, we can find an element a in A such that D(a, B_X) ∩ A = {a}. Utilize the separating hyperplane theorem or the geometry of the Banach space to show this intersection property.

Understand the Drop Property for part (ii)

A Banach space X has the drop property if for any closed set S such that S ∩ B_X is empty, there exists s in S such that D(s, B_X) ∩ S = {s}. The goal is to link this property with X being reflexive and having the Kadec property.

Prove one direction of part (ii)

Assume X is reflexive and has the Kadec property. Show that for any closed set S with S ∩ B_X = ∅, the drop property holds. Since X is reflexive, weak compactness and the Kadec property will be used to prove that D(s, B_X) ∩ S is a singleton {s}.

Prove the converse direction of part (ii)

Assume X has the drop property. Show that X must be reflexive and that norm and weak convergence in the unit sphere coincide. Use properties of convexity, intersections, and weak compactness to establish reflexivity and the Kadec property.

Key concepts

Banach Space

A Banach space is a vector space complete with respect to a norm. This means that every Cauchy sequence (a sequence where terms get arbitrarily close to each other) has a limit within the space.

Banach spaces are important in various areas like functional analysis, where they help in understanding the behavior of different mathematical functions and operators. Key properties include:

• Norm Completeness: Every sequence that appears to converge to a point must actually have a limit in the space.
• Linear Structure: Banach spaces are vector spaces, meaning you can add any two elements and multiply them by scalars.

Example: The space of continuous functions over a closed interval with the maximum norm is a Banach space.
Understanding Banach spaces is crucial for advanced studies involving functional analysis and spaces of functions.

Drop Property

The drop property in a Banach space is a geometric condition related to distance and convex hulls.
Let's break this down:

• Drop Definition: For any element outside the unit ball, the drop is the convex hull of that element and the unit ball.
• Formal Definition: Given a closed set S that does not intersect the unit ball, there exists an element in S such that its drop (with respect to the unit ball) intersects S only trivially.

This property helps in understanding the structure and geometry of Banach spaces. It ensures that in certain configurations, there is an isolated element with a specific interaction pattern with the unit ball.
For example, in proving the drop property, you often rely on concepts like the separating hyperplane theorem, showing geometric separations in the space.

Reflexivity

Reflexivity is a vital concept in Banach spaces, indicating a kind of 'symmetry' in dual spaces.

A Banach space X is reflexive if it is naturally isomorphic to its double dual, meaning there's a natural, intuitive correspondence between the space and its dual-dual.
Key takeaways include:

• Characterization: For reflexive spaces, every bounded sequence has a weakly convergent subsequence.
• Implications: Reflexivity often implies strong geometric and functional properties, aiding in solving equations and optimization problems.

Reflexivity is tied closely with the Kadec property and the drop property. When X has reflexivity, certain geometric properties like compactness in the weak topology follow.
This is crucial when proving equivalences between the drop property and other properties in a Banach space.

Kadec Property

The Kadec property offers insights into how sequences behave in Banach spaces. Specifically, it links the norm and weak convergences.

A Banach space has the Kadec property if, within its unit sphere, every sequence that converges weakly also converges in norm.
Here are the essentials:

• Norm-Weak Convergence: In simple terms, sequences converge not only in a 'loose' sense but also in a strict norm sense.
• Utility: Helps in establishing equivalence between different kinds of limits, which is useful for functional analysis and optimization.

Example: Hilbert spaces (a type of Banach space) usually have the Kadec property.
Understanding this property is crucial when examining the interaction between different types of convergences in Banach spaces and proving theorems that require these convergence relationships.

Convex Hull

The convex hull is a fundamental concept in geometry, especially within Banach spaces.

The convex hull of a set is the smallest convex set containing that set. For a set of points, it's the 'tightest' convex boundary that encloses all these points.
Key points include:

• Definition: Convex hull of a set S, denoted conv(S), is all convex combinations of points in S.
• Computational Importance: Convex hulls are crucial in optimization, providing necessary structures for linear programming and more.

In the context of the drop property, the drop is defined as the convex hull of the unit ball and a specific element.
This concept is widely used in different areas, from optimization and computational geometry to the analysis of Banach spaces and convex analysis.