Proving The Non-Reflexivity Of Quotient Space ℓ∞/c₀ A Detailed Explanation

Introduction

In the realm of functional analysis, understanding the properties of Banach spaces is crucial. Reflexivity, a key characteristic, plays a significant role in various theoretical and applied contexts. This article delves into the fascinating topic of proving that the quotient space ℓ^∞/c₀, constructed from the space of bounded sequences (ℓ^∞) and the space of sequences converging to zero (c₀), is not reflexive. The exploration involves delving into the definitions of ℓ^∞ and c₀, understanding quotient spaces, and leveraging concepts from duality theory to demonstrate the non-reflexivity. This discussion aims to provide a comprehensive understanding of the non-reflexivity of ℓ^∞/c₀, a result that has implications in functional analysis and related fields. The concepts of normed spaces, dual spaces, and the intricacies of reflexivity are central to this analysis. We will navigate through these concepts, providing clear explanations and building a rigorous argument to support our claim. This article is structured to benefit both students learning functional analysis and researchers interested in deeper properties of Banach spaces. Our goal is to provide not only a proof but also a conceptual understanding of the underlying principles.

Defining ℓ∞ and c₀

To embark on our journey, let's first define the spaces ℓ^∞ and c₀ with precision. The space ℓ^∞, which is at the heart of our discussion, represents the vector space encompassing all bounded sequences of real (or complex) numbers. A sequence x = (x₁, x₂, x₃, ...) belongs to ℓ^∞ if there exists a finite upper bound M such that |x_n| ≤ M for all n. The norm associated with ℓ^∞ is the supremum norm, defined as ||x||_∞ = sup_n |x_n|. This norm quantifies the “size” or “magnitude” of a bounded sequence. The space ℓ^∞ equipped with this norm forms a Banach space, a complete normed vector space, which is a cornerstone in functional analysis. Understanding the completeness of ℓ^∞ is vital, as it ensures that Cauchy sequences within the space converge to a limit that is also within the space. This property is essential for many analytical arguments. Now, let's turn our attention to c₀. The space c₀ is a subspace of ℓ^∞, consisting of all sequences that converge to zero. Formally, a sequence x = (x₁, x₂, x₃, ...) belongs to c₀ if lim_n→∞ x_n = 0. The norm on c₀ is inherited from ℓ^∞, which is the supremum norm. Similar to ℓ^∞, c₀ is also a Banach space, meaning it is complete under the supremum norm. The relationship between ℓ^∞ and c₀ is crucial. c₀ is a closed subspace of ℓ^∞, a fact that is important when considering quotient spaces. The closedness of c₀ in ℓ^∞ ensures that the quotient space ℓ^∞/c₀ is well-defined and inherits desirable properties from the parent spaces. The sequences in c₀ gradually “fade away” as n increases, a characteristic that distinguishes them from general bounded sequences in ℓ^∞.

Understanding Quotient Spaces and Reflexivity

Having defined ℓ^∞ and c₀, we now shift our focus to the concept of quotient spaces and reflexivity, essential ingredients in our quest to demonstrate that ℓ^∞/c₀ is not reflexive. A quotient space, in general, is a vector space formed by “dividing” a vector space by one of its subspaces. More formally, given a normed space X and a closed subspace Y, the quotient space X/Y is the set of all cosets x + Y, where x ∈ X. A coset x + Y consists of all vectors in X that differ from x by a vector in Y. The norm on the quotient space X/Y is defined as ||x + Y|| = inf_y∈Y ||x - y||, representing the distance from the coset x + Y to the origin. Applying this to our specific case, the quotient space ℓ^∞/c₀ consists of cosets of the form x + c₀, where x is a bounded sequence in ℓ^∞. The norm of a coset x + c₀ in ℓ^∞/c₀ measures the distance from the sequence x to the subspace c₀. This distance captures the “essential” boundedness of x modulo sequences that converge to zero. To grasp reflexivity, we need to delve into the concept of dual spaces. The dual space X** of a normed space X is the space of all bounded linear functionals from X to the scalar field (typically real or complex numbers). A bounded linear functional is a linear map that doesn't “blow up” vectors too much, quantified by its norm. The norm of a linear functional f is defined as ||f|| = sup_||x||≤1 |f(x)|. The dual space X** itself is a Banach space under this norm. The dual space construction can be iterated, leading to the second dual X***, which is the dual of X**. There exists a natural embedding J from X into its second dual X***, called the canonical embedding. This map sends a vector x in X to the functional J(x) in X***, defined by J(x)(f) = f(x) for all f in X**. A Banach space X is said to be reflexive if this canonical embedding J is a surjective isometry, meaning it maps X onto the entire X*** while preserving norms. In simpler terms, a reflexive space is “the same” as its second dual. Reflexivity is a powerful property, implying several important consequences, such as the existence of minimizers for certain functionals and the validity of the Banach-Alaoglu theorem. However, not all Banach spaces are reflexive. Our goal is to demonstrate that ℓ^∞/c₀ falls into this category of non-reflexive spaces.

Proving the Non-Reflexivity of ℓ∞/c₀

The heart of our discussion lies in proving the non-reflexivity of the quotient space ℓ^∞/c₀. This requires a strategic approach that combines our understanding of ℓ^∞, c₀, quotient spaces, and reflexivity. The key idea is to show that the canonical embedding J from ℓ^∞/c₀ into its second dual ( ℓ^∞/c₀ )** is not surjective. This means we need to find a functional in the second dual that is not in the image of J. To achieve this, we leverage the fact that the dual of c₀ is ℓ¹ (the space of absolutely summable sequences) and the dual of ℓ¹ is ℓ^∞. This duality relationship provides a crucial link in our argument. Let's consider a bounded linear functional F on ℓ^∞/c₀. By the definition of the quotient norm, for any coset x + c₀ in ℓ^∞/c₀, we have ||x + c₀|| = inf_y∈c₀ ||x - y||_∞. This infimum represents the distance from the sequence x to the subspace c₀. Now, consider a functional φ in the second dual ( ℓ^∞/c₀ ). This φ is a bounded linear functional on the dual space of ℓ^∞/c₀. If ℓ^∞/c₀ were reflexive, then for every such φ, there would exist a coset x + c₀ in ℓ^∞/c₀ such that φ(f) = f(x + c₀) for all bounded linear functionals f on ℓ^∞/c₀. However, we will construct a φ that violates this condition. To construct such a φ, we utilize the Hahn-Banach theorem, a fundamental result in functional analysis. The Hahn-Banach theorem allows us to extend a bounded linear functional defined on a subspace to the entire space while preserving its norm. Consider the subspace c of ℓ^∞, which consists of all convergent sequences. c contains c₀ as a closed subspace. Define a linear functional L on c by L(x) = lim_n→∞ x_n, where x = (x₁, x₂, x₃, ...) is a convergent sequence. L measures the limit of the sequence x. It can be shown that ||L|| = 1. By the Hahn-Banach theorem, we can extend L to a functional L̃ on ℓ^∞ with ||L̃|| = ||L|| = 1. This L̃ is a bounded linear functional on ℓ^∞. Now, consider the quotient map q: ℓ^∞ → ℓ^∞/c₀, defined by q(x) = x + c₀. We can define a functional f on ℓ^∞/c₀ by f(x + c₀) = L̃(x). This f is a bounded linear functional on ℓ^∞/c₀. Define a functional φ in ( ℓ^∞/c₀ ) by φ(f) = 1, where f is the functional we just constructed. Suppose, for the sake of contradiction, that there exists a coset x + c₀ in ℓ^∞/c₀ such that φ(g) = g(x + c₀) for all bounded linear functionals g on ℓ^∞/c₀. Then, in particular, 1 = φ(f) = f(x + c₀) = L̃(x). This implies that L̃(x) = 1. However, we can construct a sequence y in c₀ such that ||x - y||_∞ < 1. This implies that | L̃(x - y) | ≤ || L̃ || || x - y ||_∞ < 1. Since y is in c₀, L̃(y) = 0, so | L̃(x) | < 1, which contradicts L̃(x) = 1. This contradiction demonstrates that there is no such coset x + c₀ in ℓ^∞/c₀ that represents the functional φ in the second dual. Therefore, the canonical embedding J is not surjective, and ℓ^∞/c₀ is not reflexive.

Implications and Conclusion

The non-reflexivity of ℓ^∞/c₀ has several important implications within functional analysis and related areas. Reflexivity is a desirable property for Banach spaces, as it guarantees the validity of certain theorems and simplifies various analytical arguments. Non-reflexive spaces, like ℓ^∞/c₀, exhibit more complex behavior and require more nuanced techniques for analysis. One significant implication is in the realm of optimization. In reflexive spaces, the existence of minimizers for certain functionals is often guaranteed under relatively mild conditions. However, in non-reflexive spaces, such existence results may fail, requiring stronger assumptions or alternative approaches. Another implication arises in the study of operator theory. The properties of operators acting on reflexive spaces are often simpler to analyze compared to those acting on non-reflexive spaces. The non-reflexivity of ℓ^∞/c₀ necessitates the development of specialized tools and techniques for studying operators defined on this space. Furthermore, the quotient space ℓ^∞/c₀ serves as a valuable example in the broader theory of Banach spaces. Its non-reflexivity highlights the diversity and complexity of Banach space structures. Understanding the properties of ℓ^∞/c₀ provides insights into the limitations of reflexivity and the need for alternative concepts and tools in functional analysis. In conclusion, we have rigorously demonstrated that the quotient space ℓ^∞/c₀ is not reflexive. This proof involved a careful construction of a functional in the second dual that cannot be represented by any element in the original space. The non-reflexivity of ℓ^∞/c₀ underscores the importance of understanding the nuances of Banach space theory and the limitations of certain desirable properties like reflexivity. This result has implications in various areas of functional analysis, including optimization, operator theory, and the general study of Banach space structures. The analysis presented here not only provides a concrete example of a non-reflexive space but also serves as a stepping stone for further exploration into the rich and intricate world of functional analysis.