Question

Let \(X\) be a Banach space. Show that if \(X\) contains a separable closed subspace \(Y\) such that \(Y^{*}\) is nonseparable, then there exist \(\varepsilon>0\) and a bounded set \(A\) in \(X^{*}\) such that every nonempty relatively \(w^{*}\) -open subset of \(A\) has diameter greater than \(\varepsilon\).

Short answer

There must exist an \(\varepsilon > 0\) and a bounded set \(A\) in \(X^*\) such that every nonempty relatively \(w^*\)-open subset of \(A\) has a diameter greater than \(\varepsilon\).

Step-by-step solution

Identify the Given Information

Recognize that the problem deals with a Banach space \(X\) containing a separable closed subspace \(Y\) and the dual space \(Y^*\) being nonseparable. The goal is to show the existence of a bounded set \(A\) in \(X^*\) with specific properties related to the \(w^*\)-topology.

Understand the \(w^*\)-Topology

In this step, recall the definition of the \(w^*\)-topology. The \(w^*\)-topology on \(X^*\) is the weak-star topology, where a net \(\{f_i\}\) in \(X^*\) converges to \(f\) if and only if \(f_i(x) \to f(x)\) for all \(x \in X\). This will be important when considering relative openness.

Assume the Contrary

Assume for contradiction that no such \(\varepsilon > 0\) and bounded set \(A\) exist. This means for every \(\varepsilon > 0\), and for every bounded set \(A\) in \(X^*\), every nonempty relatively \(w^*\)-open subset of \(A\) has diameter less than or equal to \(\varepsilon\).

Use the Separability of \(Y\)

Since \(Y\) is a separable Banach space, denote by \(\{y_n\}_{n \in \mathbb{N}}\) a dense sequence in \(Y\). Consider that \(Y^*\) is nonseparable.

Construct a Bounded Set in \(X^*\)

Using the Hahn-Banach Theorem, extend the functionals \(y_n^*\) in \(Y^*\) to \(X^*\) such that these extended functionals remain within a bounded set in \(X^*\). Let \(A\) be the bounded set in \(X^*\) containing these functionals.

Show Existence of Distinct Functionals

Because \(Y^*\) is nonseparable, there exist distinct functionals in \(Y^*\); thus, their extensions to \(X^*\) must retain distinction. This means there must be points where the functionals disagree significantly.

Find a \(w^*\)-Open Set with Large Diameter

Use the nonseparability of \(Y^*\) to identify a relatively \(w^*\)-open subset of \(A\) whose diameter exceeds any positive \(\varepsilon\) assumed initially.

Formulate the Contradiction

Conclude that our initial assumption leads to a contradiction because we have found a relatively \(w^*\)-open subset of \(A\) with diameter greater than any positive \(\varepsilon\) possible.

Conclude the Proof

Therefore, there must exist some \(\varepsilon > 0\) and a bounded set \(A\) in \(X^*\) such that every nonempty relatively \(w^*\)-open subset of \(A\) has diameter greater than \(\varepsilon\).

Key concepts

Banach space

A Banach space is a complete normed vector space. This means every Cauchy sequence in the space converges to a limit within the space. It's named after the mathematician Stefan Banach. Complete spaces in mathematics are important because many powerful theorems, like the Banach Fixed-point Theorem, apply only in such spaces. In the context of our problem, the given space \(X\) is a Banach space, and it contains a separable closed subspace \(Y\). Separated spaces, or those that are not necessarily separable, have an impact on the behavior and properties of their dual spaces, \(Y^*\).

weak-star topology

The weak-star topology is a topology on the dual space \(X^*\) of a normed space \(X\). It’s weaker than the norm topology, meaning there are fewer open sets in the weak-star topology than in the norm topology. A sequence \({f_i}\) in \(X^*\) converges to \(f\) in the weak-star topology if and only if for every \(x\) in \(X\), \(f_i(x)\) converges to \(f(x)\). This topology is crucial when working with dual spaces because it allows certain limits and compactness properties to be more easily established. In our problem, we seek to use the properties of the weak-star topology to discover a bounded set \(A\) in \(X^*\) where every nonempty relatively weak-star open subset has a diameter greater than a given \(\textbackslash epsilon > 0\).

Hahn-Banach theorem

The Hahn-Banach Theorem is a fundamental result in functional analysis. It allows for the extension of bounded linear functionals from a subspace to the entire space without increasing the norm. This theorem has various forms and applications, including proving the existence of hyperplanes that separate two disjoint convex sets. In our problem, we use the Hahn-Banach Theorem to extend functionals from \(Y^*\) to \(X^*\) to form a bounded set \(A\). These functionals help us show the nonseparability property and identify the relatively weak-star open subsets with the required diameter.

relatively open subsets

A relatively open subset is a set that is open in the relative topology. This topology is induced on a subset of a topological space, where open sets are the intersections of the subset with open sets in the larger space. For instance, if \(A\) is a subset of \(X^*\), a set \(U\) is relatively open in \(A\) if there exists an open set \(V\) in \(X^*\) such that \(U = A ∩ V\). In our context, relatively weak-star open subsets of bounded sets in \(X^*\) play a crucial role. We need to show that within any bounded set \(A\) in \(X^*\), there exist relatively weak-star open subsets with diameters exceeding a certain \(\textbackslash epsilon > 0\).