Question

Let \(X, Y\) be Banach spaces and \(T \in \mathcal{B}(X, Y)\). Show that if \(Y\) is separable and \(T\) is onto \(Y\), then there is a separable closed subspace \(Z\) of \(X\) such that \(T(Z)=Y\).

Short answer

There exists a separable closed subspace Z of X such that \(T(Z) = Y\).

Step-by-step solution

Understand the given conditions

We are given two Banach spaces, X and Y, and a bounded linear operator T from X to Y, denoted as \(T \in \mathcal{B}(X, Y)\). Y is separable, and T is onto, meaning T(X) = Y.

Define separability

Since Y is separable, there exists a countable dense subset \(D = \{y_n\}_{n \in \mathbb{N}}\) in Y. This means every element of Y can be approximated arbitrarily closely by elements of D.

Preimage of the countable dense subset

For each \(y_n \in D\), choose \(x_n \in X\) such that \(T(x_n) = y_n\). The set \(\{x_n\}_{n \in \mathbb{N}}\) is a countable subset of X.

Subspace generated by the preimage

Consider the closed subspace Z generated by the set \(\{x_n\}_{n \in \mathbb{N}}\). This subspace Z is countable and closed, hence separable since a countable set in a metric space is separable.

Image of Z under T

Since T is linear and onto, we check \(T(Z)\). Because Z contains preimages of the countable dense subset D in Y, the image of Z under T will be dense in Y. Since Y is already assumed to be separable and the image of a dense subspace under a bounded linear operator is also dense, \(T(Z)\) must be equal to Y.

Conclusion

We have constructed a separable closed subspace Z of X such that \(T(Z) = Y\), as required.

Key concepts

Bounded Linear Operator

Understanding bounded linear operators is crucial in functional analysis. These operators map one Banach space to another while preserving both linearity and the boundedness property. Formally, a linear operator T from Banach space X to Banach space Y is called bounded if there exists a constant C such that for every x in X, we have:\[ \|T(x)\|_Y \leq C \|x\|_X \]
This means the operator does not magnify the input more than a constant multiple.
Some key characteristics of bounded linear operators include:

• Continuity: A bounded linear operator is continuous.
• Norm: The smallest such constant C is called the operator norm, denoted by \( \|T\| \).
• Stability: These operators maintain their properties even under limit operations.

In our exercise, T is a bounded linear operator between Banach spaces X and Y, ensuring it is both continuous and satisfies the boundedness property.

Dense Subset

To grasp the idea of a dense subset, think about how closely elements of a set can approximate other elements in a space. A subset D of a Banach space Y is dense if every element of Y can be approximated arbitrarily well by elements of D.
Mathematically, D is dense in Y if for every \(y \in Y\) and every \(\epsilon > 0\), there exists a \(d \in D\) such that \( \|y - d\| < \epsilon \).
In our exercise, because Y is separable, it contains a countable dense subset D, meaning there's a sequence \(\{y_n\}_{n \in \mathbb{N}}\) in Y such that any element in Y is the limit of a sequence from D.

• A dense subset spans the entire space in the sense of limits.
• Countable dense subsets demonstrate the separability of space.
• In our context, the dense subset of Y helps construct the subspace of X that maps onto Y.

This concept is pivotal in the exercise, as it allows us to use the preimages of elements in D to form the required subspace Z in X.

Preimage

The preimage of a set under a function is the collection of all elements in the domain that map to elements of the set in the codomain. Formally, for a function \(T : X \rightarrow Y\) and a subset \(A \subseteq Y\), the preimage \(T^{-1}(A)\) is given by:
\[ T^{-1}(A) = \{ x \in X \mid T(x) \in A \} \]
In our exercise, we select preimages of elements of the dense subset D in Y. Specifically, for each \(y_n \in D\), there exists \(x_n \in X\) such that \(T(x_n) = y_n\). This set \(\{x_n\}_{n \in \mathbb{N}}\) is a countable subset in X.
Using these preimages, we can generate a subspace Z of X which captures the dense properties of D in Y:

• Preimages connect elements between the domain and codomain.
• They help in constructing meaningful subsets in the domain that map onto particular subsets in the codomain.
• The set \(\{x_n\}_{n \in \mathbb{N}}\) ensures that the image of Z under T is dense in Y, fulfilling the exercise's requirement.

Overall, understanding preimages helps us bridge the properties of dense subsets between the given Banach spaces.