Question

Let \(C\) be a weakly closed set in a Banach space \(X\). Assume that every continuous linear functional on \(X\) attains its supremum over \(C .\) Is \(C\) weakly compact? Hint: Yes see the proof of Theorem \(3.58\)

Short answer

Theorem 3.58 guarantees that C is weakly compact since every continuous linear functional on X attains its supremum over C.

Step-by-step solution

Understand Definitions

A set is weakly closed if it contains all its limit points with respect to the weak topology. A set is weakly compact if it is both bounded and weakly closed, and every sequence has a weakly convergent subsequence whose limit is contained in the set.

Recognize Hypotheses

The given hypothesis states that every continuous linear functional on the Banach space reaches its supremum over the set C. This property is crucial for showing weak compactness.

Use Theorem 3.58

Theorem 3.58 indicates that if every continuous linear functional attains its supremum over a set, then that set is weakly compact. Consider how this theorem directly applies to the given set C.

Conclusion

Since C satisfies the condition from Theorem 3.58, C must be weakly compact.

Key concepts

Weak Topology

The weak topology on a Banach space is generated by continuous linear functionals. Unlike the norm topology, it is typically coarser, meaning it has fewer open sets. Weak topology focuses on convergences defined by these functionals.
When we say a sequence \(x_n\) converges to \(x\) weakly, it means for every continuous linear functional \(f\), the sequence \(f(x_n)\) converges to \(f(x)\).
This makes the weak topology particularly important when dealing with bounded sets or studying properties such as weak compactness.
Weak topology ensures that certain functional limits are preserved, even if the norms are not.

Banach Space

A Banach space is a vector space equipped with a norm, where every Cauchy sequence converges within the space. This completeness is a key feature.
Typical examples include spaces like \(L^p\) spaces and \(C([a, b])\), spaces of continuous functions over closed intervals.
These spaces allow the use of several functional analysis techniques, such as the Banach-Alaoglu theorem and the Hahn-Banach theorem.
When analyzing weak topologies and properties such as weak compactness, working within a Banach space provides necessary structure and tools.

Continuous Linear Functional

Continuous linear functionals are mappings from a Banach space to the field of real or complex numbers that preserve vector addition and scalar multiplication.
These functionals are central to defining the weak topology.
For instance, if we have a functional \(f\) and sequences \(x_n\) in our Banach space, the weak convergence \(x_n \to x\) implies \(f(x_n) \to f(x)\).
Further, the dual space of a Banach space is formed by all continuous linear functionals on that space, creating a bridge between various functional analysis concepts.
In problems like the exercise discussed, these functionals help establish properties such as weak compactness by checking limits and suprema.

Weakly Closed Set

A set in a Banach space is weakly closed if it contains every limit point of all weakly convergent sequences within the set.
This property is critical when discussing weak compactness.
Such sets are closed with respect to the weak topology, though they might not be closed in the norm topology.
In our exercise, the weakly closed property, combined with the conditions on continuous linear functionals, leads directly to the conclusion of weak compactness through Theorem 3.58.
In essence, weakly closed sets ensure the preservation of weak limits, making them integral in the study of functional analysis.