Question

The Josefson-Nissenzweig theorem asserts that, for every Banach space \(X\), there is a sequence \(\left\\{f_{n}\right\\} \subset S_{X^{*}}\) such that \(f_{n} \stackrel{w^{*}}{\rightarrow} 0\) ([Dis2]). Show that, given \(f \in B_{X^{*}}\), there is a sequence \(\left\\{f_{n}\right\\} \subset S_{X^{*}}\) such that \(f_{n} \stackrel{w^{*}}{\rightarrow} f\)

Short answer

By adding a small perturbation from a sequence provided by the Josefson-Nissenzweig theorem, a sequence \( f_n \subset S_{X^*} \) can be constructed such that \( f_n \stackrel{w^*}{\rightarrow} f \).

Step-by-step solution

- Understand the Definitions

Recall the following definitions: a Banach space is a complete normed vector space. The dual space, denoted as \( X^* \), is the space of all continuous linear functionals on \( X \). Moreover, the weak-* topology on \( X^* \) is the weakest topology that makes all evaluations at points of \( X \) continuous.

- Recognize the Result of Josefson-Nissenzweig Theorem

The theorem states that for any Banach space \( X \), there exists a sequence \( \{ f_n \} \subset S_{X^*} \) such that \( f_n \stackrel{w^*}{\rightarrow} 0 \). Here, \( S_{X^*} \) denotes the unit sphere in the dual space \( X^* \).

- Shift the Sequence to Approach \( f \)

Given any functional \( f \in B_{X^*} \) (the closed unit ball of \( X^* \)), consider the sequence \( \{ g_n \} \subset S_{X^*} \) provided by the Josefson-Nissenzweig theorem such that \( g_n \stackrel{w^*}{\rightarrow} 0 \). Define a new sequence \( f_n = f + \epsilon_n g_n \), where \( \epsilon_n \) is chosen such that \( 0 < \epsilon_n \to 0 \).

- Verify the Conditions of the Sequence

Confirm that each \( f_n \in S_{X^*} \) by ensuring the norm condition \( \| f_n \| = 1 \) holds: due to the properties of the norm in Banach spaces and the requirement that \( \epsilon_n \) be sufficiently small, it can be shown that \( \| f + \epsilon_n g_n \| \leq 1 \).

- Establish Weak-* Convergence

Finally, show that \( f_n \stackrel{w^*}{\rightarrow} f \). For each \( x \in X \) and the sequence \( \{ f_n \} \), compute \( f_n(x) = f(x) + \epsilon_n g_n(x) \). As \( g_n \stackrel{w^*}{\rightarrow} 0 \) implies \( g_n(x) \rightarrow 0 \), and \( \epsilon_n \to 0 \), it follows that \( f_n(x) \to f(x) \).

Key concepts

Banach Space

In mathematics, a Banach space is a type of vector space that is both complete and normed. To break these terms down:
Complete refers to the idea that every Cauchy sequence (a sequence where the elements get arbitrarily close to each other) in the space must converge to a point within the space.
Normed indicates that the space is equipped with a function called a norm, which assigns a non-negative length or size to each vector in the space.

Banach spaces are pivotal in functional analysis because they provide the perfect blend of algebraic structure and topological properties. They help us handle infinite-dimensional vector spaces in a controlled manner, making them extremely useful in fields such as differential equations and quantum mechanics.

Weak-* Topology

The weak-* topology is a specific type of topology that can be placed on the dual space of a Banach space. Here's what you need to know:
The dual space, denoted as \(X^*\), consists of all continuous linear functionals on the Banach space \(X\). A functional is essentially a map that takes a vector in \(X\) and returns a scalar.
The weak-* topology is the weakest (or coarsest) topology on \(X^*\) that still makes all the functionals evaluate to a continuous map when applied to any point in \(X\).

This topology is less fine than the norm topology, meaning fewer sets are open in the weak-* topology than in the norm topology. However, it is particularly useful when working with sequences in the dual space, as it allows for the convergence of these sequences in ways that are compatible with the constraints of functional analysis.

Dual Space

The dual space, or \(X^*\), of a Banach space \(X\), consists of all linear functionals on \(X\). To help you understand:
A functional is like a function, but instead of taking vectors and producing other vectors, it takes vectors and produces scalars.
Linear means that the functional must satisfy specific properties: it respects both vector addition and scalar multiplication.

The dual space is crucial because it provides a way to 'measure' vectors. The dual space gives us tools to investigate and manipulate Banach spaces effectively. For example, in the Josefson-Nissenzweig theorem, the sequence of functionals \( \{ f_n \} \) in the dual space is used to explore deeper properties of the Banach space itself. This functional perspective is invaluable in fields such as optimization and theoretical physics, where understanding how to map complex vector structures to simple scalar quantities can yield powerful insights.