Question

Let || \cdot || be an LUR norm on a separable Banach space \(X .\) Show that the set \(S\) of all elements in \(X^{*}\) that attain their norm is dense and \(G_{\delta}\) in \(X^{*}\)

Short answer

The set \( S \) of all functionals that attain their norm is dense and a \( G_{\delta} \) set in \( X^* \).

Step-by-step solution

- Understand the Problem

Given a separable Banach space \( X \) with a norm \( || \, \cdot \, || \), you need to show that the set \( S \) of all elements in \( X^* \) (the dual space of \( X \)) that attain their norm is dense and \( G_{\delta} \) (a countable intersection of open sets) in \( X^* \).

- Define the Set S

Define the set \( S = \{ f \, \in \, X^* \, : \, \exists \, x \, \in \, X \, \text{with} \, ||x|| = 1 \, \text{and} \, ||f|| = |f(x)| \} \). This is the set of all linear functionals in the dual space that attain their norm.

- Show that S is Dense in X*

Use the fact that in an LUR (Locally Uniformly Rotund) norm, for any functional \( f \, \in \, X^* \) and any \( \,\varepsilon > 0 \), there exists \( g \, \in \, S \) such that \( ||f - g|| < \varepsilon \). This is because, in LUR norms, the dual ball is norming, making \( S \) dense in \( X^* \).

- Show that S is G_{\delta} in X*

Express \( S \) as a countable intersection of open sets. For each natural number \( n \), define the set \[ S_n = \bigcup_{x \, \in \, X, \, ||x|| = 1} \{ f \, \in \, X^* \, : \, |f(x)| > ||f|| - \frac{1}{n} \} \]. Each \( S_n \) is open in \( X^* \), and \[ S = \bigcap_{n=1}^{\infty} S_n \] displaying that \( S \) is a \( G_{\delta} \) set in \( X^* \).

Key concepts

LUR norm

A Locally Uniformly Rotund (LUR) norm is an important concept in Banach space theory. An LUR norm ensures that for every sequence \( \{ x_n \} \) in the space such that \( || x_n || \rightarrow 1 \) and \( x_n \rightarrow x \) weakly, we must have \( x = 0 \) eventually. This property is stronger than simple convexity and provides better control over the geometry of the space. In our exercise, the LUR norm helps establish the density of the set of norm-attaining functionals in the dual space.

Separable Banach space

A Banach space is separable if it contains a dense countable subset. This essentially means we can approximate any element in the space arbitrarily well using elements from a countable set. Separable spaces are significant because they allow us to use properties like countable intersections in arguments.
For instance, in our problem, the separability of the Banach space \( X \) helps in discussing the properties of the set \( S \) in the dual space \( X^* \).

Dual space of Banach space

The dual space \( X^* \) of a Banach space \( X \) consists of all bounded linear functionals on \( X \). Each functional \(f \) in \( X^* \) assigns a real or complex number to every element in \( X \) such that \( f \) is linear and continuous with respect to the norm. In our exercise, we consider the set \( S \) of functionals in \( X^* \) that attain their norm. The properties of the dual space are crucial to showing that \( S \) is dense and constitutes a \( G_{\delta} \) set.

Norm-attaining functionals

A functional \( f \) in \( X^* \) is said to attain its norm if there exists an element \( x \in X \) with \( ||x|| = 1 \) such that \(|f(x)| = ||f|| \). The set \( S \) of all such functionals is central to our exercise. To show that \( S \) is dense, we leverage the LUR property. To show that \( S \) is a \( G_{\delta} \) set, we use the representation of \( S \) as a countable intersection of open sets.

Gδ sets

A \( G_{\delta} \) set is a set that can be expressed as a countable intersection of open sets. This concept comes from topology and is useful for proving that certain sets have desirable properties. In our exercise, we establish that the set \( S \) of norm-attaining functionals is \( G_{\delta} \) by showing that \(\bigcap_{n=1}^{\infty} S_n \text{, where \ S_n = \bigcup_{x \in \ X, || x || = 1} \{ f \in X^* : |f(x)| > ||f|| - \frac{1}{n} \} \}\forall n \), is a countable intersection of open sets.