Question

Let \(Y\) be a closed subspace of a Banach space \(X .\) Show that if \(Y\) and \(X / Y\) are reflexive, then \(X\) is reflexive. Thus, reflexivity is a three-space property.

Short answer

If \(Y\) and \(X/Y\) are reflexive, then \(X\) is reflexive because the natural map \(X \rightarrow X^{**}\) becomes surjective, proving the reflexivity.

Step-by-step solution

Understand Reflexivity

A Banach space is reflexive if the natural map from the space to its double dual is surjective. That is, every bounded linear functional on the dual can be represented by an element of the space itself.

Consider Given Conditions

We know that both subspace \(Y\) and the quotient space \(X/Y\) are reflexive. We need to show that this implies \(X\) is reflexive.

Use the Fact The Dual of \(X/\)Y Is Isomorphic to X Mo dulo \(Y^\bot\)

A key fact is that \((X/Y)^* eq Y^\bot\). This means $$(X/Y)^* eq Y^\bot$$ is hints isomorphic to the annihilator of \(Y\) in \(X^*\). Hence the duality plays a role.

Reflexivity Property Transfer

Given that \(Y\) is reflexive, \(Y^{**} egets Y\). Also since \(X / Y\) is reflexive \(((X/Y)^* * = X* / Y)^*= X/Y\). This is important because it means the duals share reflexivity.

Use Hahn-Banach Theorem

The Hahn-Banach theorem can give insight into reflexivity.

Show Surjectivity

We must show that the natural inclusion map \(X \rightarrow X^{**}\) is surjective. Utilizing those reflexivities and nature of closed embedded space, we can show that the map \(X \rightarrow X^{**}\) is onto.

Conclude Reflexivity

Since we have shown that \(X \rightarrow X^{**}\) is surjective, we conclude that \(X\) is reflexive, thus proving that reflexivity is a three-space property.

Key concepts

Banach spaces

Banach spaces are complete normed vector spaces. This means every Cauchy sequence in the space converges to an element within the space. Banach spaces are crucial in functional analysis because they provide a framework where many important theorems and operations are valid.
For instance:

• Given any continuous linear map from a Banach space to another, the image is also a Banach space if the map is surjective.
• Norms in Banach spaces lead to metric structures, making them akin to metric spaces with practical applications in solving differential equations.

Understanding Banach spaces helps in peering into more complex structures like reflexivity and dual spaces.

Reflexivity

A Banach space is reflexive if the natural embedding into its double dual is surjective. This means every bounded linear functional on the dual can be identified with an element of the space itself. Reflexivity implies that:

• The weak topology and weak* topology on the dual space coincide.
• Any bounded sequence in a reflexive Banach space has a weakly convergent subsequence.

Reflexivity is significant because it simplifies the analysis in the topic of functional analysis, enabling practitioners to use the properties of sequences and limits efficiently.

Hahn-Banach theorem

The Hahn-Banach theorem is a cornerstone in functional analysis. It allows us to extend bounded linear functionals from a subspace to the whole space while preserving their norm. This theorem is used to demonstrate many other fundamental results:

• It shows the existence of supporting hyperplanes in convex analysis.
• It extends functionals, allowing the study of dual spaces more comprehensively.

By leveraging the Hahn-Banach theorem, one can prove the reflexivity of spaces and solve other extension problems in Banach spaces.

Quotient spaces

Quotient spaces form a core concept in analyzing Banach spaces. Given a Banach space \(X\) and a closed subspace \(Y\), the quotient space \(X/Y\) is the set of cosets of \(Y\) in \(X\), equipped with a norm derived from \(X\). Important properties include:

• \(X/Y\) is also a Banach space.
• The dual of \(X/Y\) relates to the annihilator of \(Y\) in the dual space \(X^*\). This connection is crucial in examining reflexive properties through quotient and subspaces.

Understanding quotient spaces enables deeper insights into the structure and behavior of functional spaces.

Dual spaces

The dual space of a Banach space \(X\) (denoted as \(X^*\)) consists of all bounded linear functionals on \(X\). This space is vital because it facilitates:

• Analyzing continuous linear functionals and operator theory.
• Understanding the weak and weak* topologies, influencing convergence and compactness properties in Banach spaces.

The double dual space, \(X^{**}\), becomes significant in discussions about reflexivity, particularly. Reflexivity equates to the natural embedding of \(X\) into \(X^{**}\) being surjective, highlighting the profound relationship between \(X\) and its dual spaces.