Question

Let \(X\) be a Banach space. We say that \(X\) has the Kadec-Klee property if the weak and norm topologies coincide on \(S_{X}\). We say that \(X^{*}\) has the \(w^{*}\) -Kadec-Klee property if the \(w^{*}\) - and norm topologies coincide on \(S_{X^{*}}\) (i) Let \(X\) be a locally uniformly rotund space. Show that \(X\) has the KadecKlee property. (ii) Let \(X\) be a Banach space such that the dual norm of \(X^{*}\) is LUR. Show that \(X^{*}\) has the \(w^{*}\) -Kadec-Klee property.

Short answer

Locally uniformly rotund spaces have the Kadec-Klee property. If the dual norm of a Banach space is LUR, its dual space has the \(w^*\)-Kadec-Klee property.

Step-by-step solution

Understanding Local Uniform Rotundity (LUR)

A Banach space is locally uniformly rotund (LUR) if for every sequence \((x_n)\) in \S_X\ such that \x_n \to x\ weakly and \|x_n| \to \|x|\, we have \x_n \to x\ in norm.

Show LUR implies the Kadec-Klee Property

Suppose \X\ is LUR. Given a sequence \((x_n)\) in \S_X\ such that \x_n \to x\ weakly and \|x_n| \to \|x|\. By definition of LUR, this implies \x_n \to x\ in norm. Thus, the weak and norm topologies coincide on \S_X\, showing that \X\ has the Kadec-Klee property.

Understanding Dual Norm and \ w^* -\ Kadec-\ Klee \ Property

The dual norm on \(X^*\) means that we are considering the norm in the space of bounded linear functionals on \(X\). If \(X^*\) is LUR, then for any sequence \((f_n)\) in \(S_{X^*}\) such that \(f_n \to f\) in the weak\(*\) topology \(w^*\) and \(\|f_n\| \to \|f\|\), necessarily \(f_n\) converges to \(f\) in norm.

Show Dual Norm LUR implies \ w^* -\ Kadec-\ Klee Property

Given \((f_n)\) in \(S_{X^*}\) with \(f_n \to f\) in \(w^*\)-topology and \(\|f_n\| \to \|f\|\), since \(X^*\) is LUR, we get \(f_n \to f\) in norm. Thus, the \(w^*\) and norm topologies coincide on \(S_{X^*}\), showing \(X^*\) has the \(w^*\)-Kadec-Klee property.

Key concepts

Banach Space

A Banach space is a type of vector space equipped with a norm, where every Cauchy sequence converges in norm to a limit within the space. This means that the space is complete.
Complete means that sequences that get arbitrarily close to each other actually have a limit within the space. This property is essential in many areas of analysis.
Banach spaces are vital because they allow us to generalize many concepts from finite-dimensional linear algebra to infinite-dimensional settings.
For example, spaces of continuous functions are often Banach spaces.

Local Uniform Rotundity

Local Uniform Rotundity (LUR) is a property of a Banach space where the norm behaves especially nicely regarding limits of sequences.
Specifically, a Banach space is LUR if for any sequence \((x_n)\) in the unit sphere such that \(x_n \to x\) weakly and \(|x_n| \to |x|\), it must be that \((x_n)\) actually converges to \(x\) in norm.
This means that weak and norm convergence align in a very strong sense.
As a result, LUR spaces have the Kadec-Klee property, where the weak and norm topologies coincide on the unit sphere.

Dual Norm

The dual norm relates to the space of bounded linear functionals on a Banach space, denoted as \(X^*\).
The dual norm measures the size of functionals in a way that mirrors the original norm on the Banach space.
Specifically, for a functional \(f \) in \(X^*\), its norm is given by: \[\|f\| = \sup\limits_{x \in X, \|x\| \leq 1} |f(x)| \].
This norm ensures that we can handle functionals in a structured manner, similar to how we handle vectors in the Banach space.
For spaces where the dual norm is locally uniformly rotund (LUR), certain convergence properties hold analogously to those in the original Banach space.

Weak Topology

The weak topology on a Banach space is a way of understanding convergence that is less stringent than norm convergence.
In the weak topology, a sequence \((x_n)\) converges to \(x\) if: for every functional \(f \) in the dual space \(X^*\), \(f(x_n) \to f(x)\).
This is a broader notion of convergence and is often easier to satisfy compared to norm convergence, where \(|x_n - x| \to 0\).
Weak topology is crucial in functional analysis for studying the behavior of sequences and their limits within Banach spaces.

Norm Topology

The norm topology on a Banach space directly relates to how we typically measure convergence.
In the norm topology, a sequence \((x_n)\) converges to \(x\) if \(|x_n - x| \to 0\).
This form of convergence is straightforward and measures the actual distance between elements in the space.
It's stronger than weak convergence because it requires that the difference in the norms themselves goes to zero.
The norm topology provides a natural way to engage with Banach spaces, making it a powerful tool in analysis.