Question

Show that if \(X\) is a separable Banach space such that its second dual norm is Gâteaux differentiable, then \(X^{*}\) is separable.

Short answer

Utilize James' theorem and reflexivity properties to conclude \(X^*\) is separable based on \(X^{**}\) being Gâteaux differentiable.

Step-by-step solution

Understand the Given Information

We have a separable Banach space, denoted as \(X\). Its second dual, \(X^{**}\), has a Gâteaux differentiable norm. We need to show that \(X^*\) (the dual space of \(X\)) is separable.

Utilize the Gâteaux Differentiability

A norm is Gâteaux differentiable at a point if the limit defining the directional derivative exists at that point. Use this property in the context of \(X^{**}\).

Apply James' Theorem

James' theorem states that if a Banach space is reflexive and its dual norm is Gâteaux differentiable, then the space is separable. Apply this theorem to \(X^{**}\).

Reflexivity Implication

Recall that if \(X\) is a Banach space and \(X^{**}\) is separable and reflexive, then \(X^*\) should also be separable. This follows from the properties of dual spaces and reflexivity.

Conclude the Separability of \(X^*\)

Since \(X^{**}\) is given to be Gâteaux differentiable, \(X^{**}\) is separable. Now, according to dual space properties, \(X^*\) must also be separable.

Key concepts

Gâteaux differentiability

The concept of Gâteaux differentiability is essential in understanding the smoothness of a norm in the context of Banach spaces. A norm is Gâteaux differentiable at a point if the directional derivative exists in every direction.
In more formal terms, if \(orm{\bullet}\) denotes the norm on a Banach space \(X\), the norm is Gâteaux differentiable at \(x eq 0\) if there exists a continuous linear functional \(f \in X^*\) such that for all \(h \in X\), the following limit exists:
\(\frac{d}{dt} orm{x + th} \bigg\bracevert_{t=0} = f(h).\)
This reflects how the norm changes as you move slightly in different directions from a point \(x\). A Gâteaux differentiable norm ensures some level of smoothness, making calculations and theoretical considerations more manageable. It is an important property in functional analysis and optimization, providing insights into the behavior of norms in infinite-dimensional spaces.

James' Theorem

James' Theorem plays a crucial role in the structure of Banach spaces.
It offers a criterion for the separability of Banach spaces using reflexivity and the smoothness of norms. The theorem states:
If a Banach space \(X\) is reflexive and its dual \(X^*\) has a Gâteaux differentiable norm, then \(X\) is separable.
Reflexivity, in this context, means that the natural embedding of \(X\) into its double dual \(X^{**}\) is surjective, i.e., \(X = X^{**}\). This property is essential because it provides a strong link between a Banach space and its dual.
By applying James' Theorem, we can infer that if the second dual \(X^{**}\) of a separable Banach space \(X\) has a Gâteaux differentiable norm, then \(X^{**}\) is separable. Consequently, this allows us to conclude the separability of the dual space \(X^*\), because separability is inherited by dual spaces under certain conditions.

Reflexivity

Reflexivity in Banach spaces is a fundamental concept that connects a Banach space to its double dual. A Banach space \(X\) is reflexive if the natural map \(J_X: X \rightarrow X^{**}\), given by \(J_X(x)(f) = f(x)\) for all \(f \in X^*\), is surjective. In simpler terms, every element of the double dual can be identified with an element of the original space \(X\).
Reflexivity has several important implications:
- A reflexive Banach space \(X\) and its dual \(X^*\) have the same separability properties.
- Reflexive spaces are always complete, meaning every Cauchy sequence converges within the space.
- If \(X\) is reflexive, so are its dual space \(X^*\) and the double dual \(X^{**}\).
In the context of the given exercise, reflexivity helps us understand the relationship between \(X\), \(X^*\), and \(X^{**}\). It underlies James' Theorem and provides the basis for arguing the separability of \(X^*\) when certain conditions are met, such as the Gâteaux differentiability of \(X^{**}\).