Question

Let \(X\) be a separable Banach space. Show that there is a sequence \(\left\\{f_{n}\right\\} \subset S_{X^{*}}\) such that \(\left\\{f_{n}\right\\}\) is separating in \(X\).

Short answer

Constructing a sequence of linear functionals \{f_n\} using separability and the Hahn-Banach theorem yields a separating sequence in \( X \).

Step-by-step solution

Recall Banach Space Properties

A Banach space is a complete normed vector space. 'Separable' means that there exists a countable dense subset within the space.

Understand Separating Sequences

A sequence \(\f\right\) is said to be separating in X if for any distinct elements \(x_1, x_2 \in X\), there exists \(f \ \in \f \) such that \( f(x_1) \eq f (x_2) \).

Use the Hahn-Banach Theorem

The Hahn-Banach theorem allows the extension of linear functionals in a way that preserves their boundedness. Use this theorem to create linear functionals that will be part of \(\f\right\)

Construct the Sequence

Construct \( \{f_n\} \subseteq S_{X^*} \), ensuring that each \(f_n\) is a linear functional that distinguishes between elements of a countable dense subset \(\{x_n\} \subseteq X \).

Show Density Implies Separation

Since \(X \) is separable, for any \(x_1, x_2 \in X \) with \(x_1 \eq x_2 \), there must exist some \( f_n \in \{\f\right\} \) such that \( f_n(x_1)eq f_n(x_2) \).

Key concepts

Banach spaces

A Banach space is a type of vector space that is both complete and normed. Completeness means that every Cauchy sequence in the space converges to an element within the space. Normed indicates that each vector has a length or magnitude defined by a norm function.

Banach spaces play a critical role in functional analysis and have applications in various fields such as quantum mechanics and differential equations. They are generalizations of more familiar spaces like Euclidean spaces, but with potentially infinite dimensions.

Separable Banach spaces have the special property of containing a countable dense subset. This means there exists a countable set within the Banach space, such that every element of the space can be arbitrarily closely approximated by elements of this countable set.

Separating sequences

Separating sequences help distinguish between different elements in a Banach space. Specifically, a sequence \((f_n)\) in the dual space \(X^*\) is called separating if, for any two distinct elements \(x_1, x_2 \in X\), there exists an \(f_n\) in the sequence such that \(f_n(x_1) eq f_n(x_2)\).

This means that no matter how close \(x_1\) and \(x_2\) are, they can be 'separated' by at least one functional in the sequence. Constructing such sequences relies on the property that the Banach space is separable and utilizing the functionals from the dual space \(X^*\). For every pair of distinct elements in the dense subset, we find a functional that acts differently on these elements. Consequently, the same holds for all elements in the space because the dense subset spreads across the entire space.

Separating sequences are pivotal in the study of Banach spaces as they aid in the precise characterization and behavior of elements in these spaces.

Hahn-Banach theorem

The Hahn-Banach theorem is a fundamental result in functional analysis. It allows the extension of bounded linear functionals from a subspace of a vector space to the entire vector space without increasing their norm.

In more technical terms, given a sublinear function \(p\) on a vector space \(V\) and a linear functional \(f\) defined on a subspace \(U\) of \(V\) such that \(f\) is dominated by \(p\), there exists an extension of \(f\) to the whole space \(V\) that still does not exceed \(p\). This article thus preserves both linearity and boundedness.

The Hahn-Banach theorem is essential in constructing separating sequences in Banach spaces. It allows us to extend linear functionals defined on dense subsets to the whole space. Therefore, we can ensure that there exists a linear functional in \(X^*\) that distinguishes between any two distinct elements. This extension property enables the creation of the required separating sequences, which is crucial for analyzing the structure and properties of Banach spaces.