Question

Prove that every closed subspace of a reflexive space is itself reflexive.

Short answer

Every closed subspace of a reflexive space is reflexive because the embedding from the subspace to its double dual is surjective.

Step-by-step solution

Define Reflexive Space

A reflexive space is a Banach space \(X\) where the natural embedding of \(X\) into its double dual \(X^{**}\) is surjective, meaning that every element of \(X^{**}\) is the image of some element of \(X\). This implies that \(X \cong X^{**}\).

Understand Closed Subspace

A subspace \(Y\) of a Banach space \(X\) is called closed if it contains all its limit points, meaning that if a sequence in \(Y\) converges to some limit in \(X\), that limit is also in \(Y\).

State the Property of Reflexive Spaces and Subspaces

In a reflexive space \(X\), every closed subspace \(Y\) is also reflexive. This property results because the dual \(Y^*\) is isometrically isomorphic to the quotient \(X^*/Y^\perp\) where \(Y^\perp\) is the annihilator of \(Y\) in \(X^*\).

Establish the Connection with the Double Dual

The reflexivity of \(Y\) means \(Y\cong Y^{**}\), due to \(Y\) being closed in \(X\). Since \(X\) is reflexive, \(X\cong X^{**}\), and similarly, we can map \(Y^{**}\) back to \(Y\) through the embeddings with \(X\) and \(X^{**}\).

Prove Surjection from \(Y\) to \(Y^{**}\)

If \(Y\) is a closed subspace of the reflexive space \(X\), we can show that the inclusion map of \(Y\) into \(X\) extended to a continuous map from \(Y^{**}\) into \(X^{**}\) is surjective. Since \(X\cong X^{**}\) and restrictions apply, the surjectiveness ensures \(Y\cong Y^{**}\).

Complete the Proof with the Final Argument

Summarizing the steps, since the natural map from \(Y\) to its double dual \(Y^{**}\) is surjective due to \(X\)'s reflexivity, and since \(Y\) is closed in \(X\), every closed subspace of a reflexive space is itself reflexive.

Key concepts

Banach Space

Understanding Banach spaces is fundamental when delving into the deeper aspects of functional analysis. To put it simply, a Banach space is a complete normed vector space. This means every Cauchy sequence within it converges to a limit that lies within the space. Picture it as a flexible yet entirely bounded surface that doesn't allow any sequence to "fall out" of it.
A key feature here is the norm, which is a function that assigns a non-negative length or size to each vector in the space. The completeness aspect ensures that the space is whole and doesn't have any 'holes' where sequences could escape. Think of it like having all your tools fitting perfectly within a toolkit, with no missing pieces.

Closed Subspace

A closed subspace is a subset of a Banach space that contains all its limit points. This means, if you're working within it and you take a sequence that seems to be heading towards some limit, that limit will surely fall within the subspace. Imagine drawing a circle and knowing that if you walk along any path within that circle, you won't suddenly find yourself stepping outside.
This property is particularly important because it guarantees the stability and predictability of limits. In mathematical analysis, dealing with closed subspaces is like having an assurance that your results will stay within expected boundaries.

Double Dual

When Banach spaces take a turn for the fascinating, we talk about the double dual. If you start with a Banach space \( X \), its dual space \( X^* \) is the collection of all continuous linear functionals on \( X \). This dual space can be further expanded to the double dual \( X^{**} \).
The double dual considers functionals of the functionals in \( X^* \). This nested layer helps explore the structure of our original space in more profound ways. Why it’s crucial: in a reflexive space, you can slide comfortably between \( X \) and \( X^{**} \) since they behave almost like mirror images under specific embeddings. This elegant feature simplifies many complex operations and shows the neat symmetry within functional structures.

Surjective Map

A surjective mapping in mathematics is quite like ensuring every possible spot is covered without any gaps. When you have a map that projects from one set to another, and every element in the target set is hit at least once, that's surjection in action.
In the context of reflexive spaces, the natural embedding from a Banach space \( X \) to its double dual \( X^{**} \) is particularly surjective. This surjectiveness effectively means a one-to-one correspondence, where each point in the double dual is neatly partnered with one in the original space. This characteristic is a backbone to proving the reflexivity condition for closed subspaces, ensuring a complete and comprehensive reach back and forth within the space's structure.