Question

Let \(Y\) be a closed subspace of a Banach space \(X .\) Show that if \(X\) is separable, then \(Y\) and \(X / Y\) are separable. Show that if \(Y\) and \(X / Y\) are separable, then \(X\) is separable. Thus, separability is a three-space property. Hint: If \(\left\\{\hat{x}_{n}\right\\}\) is a dense set in \(X / Y\) and \(\left\\{x_{n}\right\\}\) is dense in \(Y\), choosing \(y_{n} \in \hat{x}_{n}\) and considering \(\left\\{y_{n}+x_{k} ; n, k \in \mathbf{N}\right\\}\), we have a dense set in \(X\).

Short answer

If \(X\) is separable, then \(Y\) and \(X/Y\) are separable. If \(Y\) and \(X/Y\) are separable, then \(X\) is separable.

Step-by-step solution

Understand the Problem

Given a Banach space \(X\), a closed subspace \(Y\), and the quotient space \(X/Y\), the task is to show separability properties among these spaces.

Separability Definition

Recall that a space is separable if it contains a countable dense subset.

Given \(X\) is separable, show \(Y\) is separable

Since \(Y\) is a subspace of \(X\), any dense subset of \(X\) intersected with \(Y\) is dense in \(Y\). Let \(\{x_n\}\) be a countable dense subset of \(X\). Then \(\{y_n\} = \{x_n \in Y\}\) is a countable dense subset of \(Y\), showing \(Y\) is separable.

Given \(X\) is separable, show \(X/Y\) is separable

Let \(\{x_n\}\) be a dense subset of \(X\) and consider the set \(\{\hat{x}_n = x_n + Y\}\) in \(X/Y\). This set is countable and dense in \(X/Y\) because any element in \(X/Y\) can be approximated by \(\{\hat{x}_n\}\).

Given \(Y\) and \(X/Y\) are separable, show \(X\) is separable

Let \(\{x_n\}\) be a dense subset of \(Y\) and \(\{\hat{y}_n\}\) be a dense subset of \(X/Y\). For each $$, choose \(y_n\) such that \(\{y_n + Y\} = \{\hat{y}_n\}\). Consider the set \(\{x_k + y_n : k, n \in \mathbb{N}\}\). This set is countable and dense in \(X\) because it approximates every element in \(X\).

Conclusion

By establishing the separability of each space with the given conditions, we demonstrate that separability is a three-space property: if \(X\) is separable, then \(Y\) and \(X/Y\) are separable, and if \(Y\) and \(X/Y\) are separable, then \(X\) is separable.

Key concepts

Banach space

A Banach space is a special type of vector space. It is a complete normed vector space, meaning it has a norm that measures the size of its elements and it is complete with respect to this norm.
Completeness means that any Cauchy sequence of vectors in the space converges to a limit within the space itself.
This is crucial for ensuring the mathematical 'well-behaved' nature of the space during analysis.
Banach spaces form the setting for many problems in functional analysis, including the exercise you've encountered.

separability

Separability is a property of a space meaning that there exists a countable dense subset within it.
A dense subset is one where every point in the space is either in the subset or arbitrarily close to a point in it. For example, the rational numbers are a countable dense subset of the real numbers.
In terms of Banach spaces, if a Banach space is separable, it means there's a countable set whose closure is the entire space.
This property significantly simplifies the analysis of the space, as working with countable sets is generally easier than working with uncountable sets.

quotient space

A quotient space in functional analysis is formed by partitioning a larger space into equivalent classes by some equivalence relation.
For Banach spaces, if you have a Banach space X and a closed subspace Y, then X/Y is the quotient space.
This quotient space contains all the cosets of the form {x + Y} for x in X. Intuitively, you can think of each coset as a point in the quotient space.
This concept plays an important role in the exercise because it helps demonstrate the separability of the larger space Y and X/Y.

dense subset

A dense subset of a space is a subset where every element of the space is either in the dense subset or a limit point of it.
If a space is separable, it means a countable dense subset exists within it.
In practical terms, having a dense subset means we can approximate any element of the space with elements from this subset.
This is precisely what the exercise capitalizes on, showing that dense subsets in Y and the quotient space X/Y can be used to find a dense subset in X.

three-space property

The three-space property in the context of Banach spaces and separability states that if a Banach space Y and its quotient space X/Y are both separable, then the original Banach space X must also be separable.
This property is significant because it offers a way to break down a complex space into simpler parts.
If we can show separability for the parts, we can infer it for the whole.
As the exercise demonstrates, this property helps connect the separability of subspaces and quotient spaces to the separability of the entire space X.