Question

Let \(X\) be a Banach space. Show that \(B_{X^{*}}\) is \(w^{*}\) -separable if and only if \(S_{X^{*}}\) is \(w^{*}\) -separable. Note that the \(w^{*}\) -separability of \(B_{X^{*}}\) is not preserved when passing to an equivalent norm (Exercise 12.40). Also, there is a space \(X\) for which \(X^{*}\) is \(w^{*}\) -separable and no equivalent norm on \(X\) exists so that \(B_{X^{*}}\) is \(w^{*}\) -separable ([JoL1])

Short answer

The unit ball \(B_{X^*}\) in \(X^*\) is \(w^*\)-separable if and only if the unit sphere \(S_{X^*}\) is \(w^*\)-separable.

Step-by-step solution

- Definitions

Define the terms relevant to the exercise: Banach space, dual space, unit ball of the dual space, and weak-* topology. Specifically, note that for a Banach space \(X\), the dual space \(X^*\) consists of all continuous linear functionals on \(X\). The unit ball \(B_{X^*}\) in \(X^*\) is defined as \(B_{X^*} = \{ f \in X^* : \|f\| \leq 1 \}\). The unit sphere \(S_{X^*}\) in \(X^*\) is defined as \(S_{X^*} = \{ f \in X^* : \|f\| = 1 \}\). Weak-* topology involves convergence of functionals as per the weak-* topology criterion.

- Sufficient Condition

Assume that \(B_{X^*}\) is \(w^*\)-separable and show that it implies \(S_{X^*}\) is \(w^*\)-separable. Since \(S_{X^*} \subset B_{X^*}\), any countable dense subset in \(w^*\)-topology of \(B_{X^*}\) will also be dense in \(S_{X^*}\) because the intersection of a dense set with a subset is dense in that subset. Thus, a countable dense subset of \(B_{X^*}\) also serves as a countable dense subset of \(S_{X^*}\).

- Necessary Condition

Assume that \(S_{X^*}\) is \(w^*\)-separable and show that it implies \(B_{X^*}\) is \(w^*\)-separable. Given \(S_{X^*}\) is \(w^*\)-separable, there exists a countable dense subset \(D \subset S_{X^*}\). Extend \(D\) to a countable dense subset of \(B_{X^*}\) by considering the fact that any point in \(B_{X^*}\) can be approximated by a scalar multiple of a point in \(S_{X^*}\) due to the norm structure. Therefore, \(B_{X^*}\) must also be \(w^*\)-separable.

- Conclusion

Conclude that \(B_{X^*}\) is \(w^*\)-separable if and only if \(S_{X^*}\) is \(w^*\)-separable. Both the sufficient and necessary conditions have been shown.

Key concepts

Dual Space

In functional analysis, the concept of a dual space is fundamental to understanding various advanced topics. The **dual space** of a Banach space, denoted as \(X^*\), is the set of all continuous linear functionals on \(X\).
Perhaps it’s easier to think of a dual space as a place where every element (functional) tells you something about the original space, \(X\), but in a linear and continuous fashion.
This means if you have a Banach space like \(X\), you can look at it in a different,

Unit Ball

Within the dual space \(X^*\), the **unit ball**, denoted as \(B_{X^*}\), plays a pivotal role in various theoretical discussions and practical applications.
The unit ball is defined as the set of all functionals in \(X^*\) whose norm is less than or equal to one:
\[B_{X^*} = \{ f \in X^* : \|f\| \leq 1 \}\]
Another related concept is the unit sphere, \(S_{X^*}\), defined similarly but where the norm equals precisely one:
\[S_{X^*} = \{ f \in X^* : \|f\| = 1 \}\]
Understanding the unit ball and unit sphere in a dual space helps grasp important properties, like separability and compactness in different topologies.

Weak-* Topology

The **weak-* topology** (also written as \(w^*\)-topology) is essential when working with dual spaces since it defines a specific type of convergence.
For a functional sequence \(\{f_n\}\) in \(X^*\), weak-* convergence occurs if, for every \(x\) in the original space \(X\), \(f_n(x)\) converges to some \(f\)(x) as \(n\) goes to infinity.
Mathematically,
\[f_n \xrightarrow{w^*} f \text{ iff } \forall x \in X, \ f_n(x) \to f(x)\]
This type of convergence is less strict than norm convergence, making it useful in many theoretical contexts since it often ensures compactness not present with norm topology.

Separable Spaces

A **separable space** is one that contains a countable, dense subset. In simple terms, you can find a countable set whose closure is the entire space.
For instance, the space of rational numbers is dense in the real numbers, hence real numbers are separable.
For a Banach space \(X\), and its dual space \(X^*\), talking about separability often involves looking at how dense subsets behave under different topologies, like weak-*.
If \(B_{X^*}\) (the unit ball in dual space) is weak-* separable, it means it has a countable, dense subset within that topology, impacting how we can approximate functionals in meaningful ways.