Question

Let \(X\) be a Banach space and \(A \subset X^{*}\). Show that \(A\) separates the points of \(X\) if and only if \(A_{\perp}=\\{0\\}\) if and only if \(\overline{\operatorname{conv}} w^{*}(A)=X^{*}\).

Short answer

A separates points of X if and only if \(A_{\perp} = \{0\}\), which is true if and only if \( \overline{\operatorname{conv}}w^{*}(A)=X^{*} \).

Step-by-step solution

- Understand the Problem

Given a Banach space \(X\) and a subset \(A \subset X^{*}\) (the dual space), we need to show that the following three statements are equivalent: 1. \(A\) separates the points of \(X\).2. \(A_{\backslash{\perp\}}=\{0\}\).3. \(\overline{\operatorname{conv}}w^{*}(A)=X^{*}\).

- Define separation of points

A set \(A \subset X^{*}\) separates the points of \(X\) if for any two distinct points \(x, y \in X\), there exists \(f \in A\) such that \(f(x) \eq f(y)\). In other words, for each \(x \in X\) with \(x \eq 0\), there exists \(f \in A\) such that \(f(x) \eq 0\).

- Define \(A_{\perp}\)

\(A_{\perp}\) is defined as the set of all \(x \in X\) such that \(f(x) = 0\) for all \(f \in A\). In symbols, \(A_{\perp} = \{ x \in X : f(x) = 0 \text{ for all } f \in A \}\).

- Express the first equivalence

If \(A\) separates the points of \(X\), then for any \(x \in X\) with \(x \eq 0\), there exists \(f \in A\) such that \(f(x) \eq 0\). This means that no non-zero \(x\) can result in zero for all \(f \in A\), thus \(A_{\perp} = \{0\}\).

- Express the second equivalence

Conversely, if \(A_{\perp} = \{0\}\), then for any \(x \in X\) with \(x \eq 0\), there must be some \(f \in A\) such that \(f(x) \eq 0\). Therefore, \(A\) separates the points of \(X\).

- Define the convex hull

The convex hull of a set \(A\) in \(X^{*}\), denoted \(\operatorname{conv}(A)\), is the set of all finite convex combinations of elements of \(A\). The weak-* closure of the convex hull, denoted \(\overline{\operatorname{conv}}w^{*}(A)\), is the closure in the weak-* topology.

- Equivalence with the convex hull

By the Hahn-Banach separation theorem, the condition \(A\) separating the points of \(X\) is equivalent to \(\overline{\operatorname{conv}}w^{*}(A)=X^{*}\). This is because every element of \(X^{*}\) can be approximated in the weak-* topology by a net in the convex hull of \(A\).

- Conclude the argument

Putting it all together, \(A\) separates the points of \(X\) if and only if \(A_{\perp}=\{0\}\), which happens if and only if \(\overline{\operatorname{conv}}w^{*}(A) = X^{*}\).

Key concepts

Separating Points

In functional analysis, the idea of separation plays a crucial role. When we say a subset \(A\) of the dual space \(X^*\) separates the points of a Banach space \(X\), we mean that for any two distinct points \(x eq y\) in \(X\), there exists at least one linear functional \(f\) in \(A\) such that \(f(x) eq f(y)\). This essentially means that these functionals can 'tell apart' different points in \(X\).

Let's break it down:

• Imagine you have two different vectors in your Banach space \(X\).
• If \(A\) separates the points of \(X\), at least one functional \(f\) in \(A\) will give different values for these two vectors.
• Mathematically: For each \(x eq 0\) in \(X\), there exists \(f eq 0\) in \(A\) such that \(f(x) eq 0\).

This concept is important when working with dual spaces because it ensures that the set \(A\) is rich enough to distinguish different points in the Banach space, making \(A\) a very useful tool for analysis.

Understanding Banach Space

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

Some key features of Banach spaces include:

• **Normed Vector Space**: Every Banach space is a normed vector space, meaning it has a norm function, usually denoted as \(||x||\), which assigns a length to each vector \(x\).
• **Completeness**: A critical property, where you can take any sequence of vectors that is getting closer together (Cauchy sequence), and their limit will also be in the space.
• **Dual Space**: Each Banach space \(X\) has a dual space \(X^*\) consisting of all continuous linear functionals on \(X\).

Banach spaces are extensive in many mathematical fields, especially in functional analysis, since they provide a controlled environment where limits and linear functions behave well.

Convex Hull

The convex hull of a set, particularly in the context of functional analysis and Banach spaces, is the smallest convex set that contains a given set. For a set \(A \subset X\*\), its convex hull consists of all convex combinations of points in \(A\). Here's what you need to know:

• **Convex Combination**: A point is a convex combination of elements from \(A\) if it can be written as $$ \sum \lambda_i f_i $$ where \(f_i \in A\), \(\lambda_i \geq 0\), and $$ \sum \lambda_i = 1 $$.
• **Weak-* Topology**: This is a topology on \(X\*\) that's coarser than the norm topology, meaning it has fewer open sets. The weak-* closure of the convex hull (denoted as $$ \overline{\operatorname{conv}}w^*(A) $$) is the set of all limits of nets of convex combinations of elements from \(A\).

Combining these concepts, when we say that $$ \overline{\operatorname{conv}}w^*(A)=X\* $$, it means that \(A\)'s convex hull, closed under the weak-* topology, fills up the entire dual space \(X\*\). This indicates that \(A\) is large and diverse enough to approximate any functional in \(X\*\).

Understanding these concepts and their interplay can deepen your grasp of how functional structures work in Banach spaces and why these properties matter in advanced mathematical analysis.