Question

Suppose a Banach space \(X\) contains an isomorphic copy of \(\ell_{1}\). Show that then \(X^{*}\) contains an isomorphic copy of \(\ell_{1}\left(2^{\mathbf{N}}\right)\)

Short answer

If \(X\) contains an isomorphic copy of \(\ell_{1}\), then \(X^*\) contains an isomorphic copy of \(\ell_{1}(2^{\mathbf{N}})\) because \(X^*\) must then contain \(\ell_{\infty}\), isometrically isomorphic to the dual of \(\ell_{1}(2^{\mathbf{N}})\).

Step-by-step solution

- Understand the Problem

We need to show that if a Banach space \(X\) contains an isomorphic copy of \(\ell_{1}\), then its dual space \(X^*\) contains an isomorphic copy of \(\ell_{1}(2^{\mathbf{N}})\).

- Identify Key Properties

Recall that a Banach space containing an isomorphic copy of \(\ell_{1}\) implies certain properties for the dual space and the sequences within it.

- Use Known Theorems

Utilize a known theorem: if \(X\) contains an isomorphic copy of \(\ell_{1}\), then \(X^*\) must contain an isomorphic copy of \(\ell_{\infty}\). This is because the dual of \(\ell_{1}\) is \(\ell_{\infty}\).

- Map to \(\ell_{\infty}\) Space

Identify or construct a sequence in \(X\) that maps to a bounded sequence in \(X^*\). Specifically, we seek a bounded linear operator from \(X\) to \(\ell_{\infty}\).

- Establish the Isomorphism

Construct the isomorphism from \(X^*\) to \(\ell_{1}(2^{\mathbf{N}})\). Utilize the fact that \( \ell_{\infty} \cong \ell_{1}(2^{\mathbf{N}})^* \) (duality of sequence spaces).

- Conclusion

Combine the results to conclude that if \(X\) contains an isomorphic copy of \(\ell_{1}\), then \(X^*\) inherently contains an isomorphic copy of \(\ell_{1}(2^{\mathbf{N}})\) due to the nature of the dual space and isomorphisms between sequence spaces.

Key concepts

isomorphic copy

An isomorphic copy of a space entails creating a version of one mathematical space within another, maintaining structural properties. Think of it as a 'clone' or a 'reproduction' of a space inside another, but still being essentially the same in terms of structure and function.
Here's an important aspect: when we say a Banach space contains an isomorphic copy of another space (like \(\backslash ell_1 \)), it means there is a sub-space within the Banach space that behaves exactly like \(\backslash ell_1 \) in terms of topology and linear structure.
So in plain terms, if our Banach space X has an isomorphic copy of \(\backslash ell_1 \), we can find within X a set that follows the same rules and properties as \(\backslash ell_1 \). Establishing such properties helps mathematicians understand how complex these spaces are by identifying simpler structures within them.

dual space

A dual space of a given space is essentially a space of all linear functionals—that is, linear maps from the original space into the field of real or complex numbers. This can also be visualized as 'super-functions' which take functions as their input and return a number. For example, if our original space is a Banach space denoted as X, its dual space is denoted as X*. The elements of X* are linear functionals.
Why is this concept crucial? Given that a linear functional is continuous, it helps us analyze and understand the original space better. When we talk about X containing an isomorphic copy of \(\backslash ell_1 \), it informs us about the richness and complexity of X. Consequently, the dual space X* inherits these properties.
In essence, if X includes \(\backslash ell_1 \), X* contains an isomorphic copy of \(\backslash ell_{\backslash infty} \), ultimately leading to containing \(\backslash ell_1(2^\mathbf{N}) \) due to the duality relationships between sequence spaces.

sequence spaces

Sequence spaces are collections of infinite sequences of numbers (such as integers or real numbers) with specific properties and structures. Examples of well-known sequence spaces include \(\backslash ell_1 \), \(\backslash ell_2 \), and \(\backslash ell_{\backslash infty} \). These spaces have different rules about convergence and size of sequences.
For instance:

• The space \(\backslash ell_1 \) consists of all sequences whose series sum is finite.
• The space \(\backslash ell_{\backslash infty} \) contains all bounded sequences.

In our specific problem, the spaces \(\backslash ell_1 \) and \(\backslash ell_{\backslash infty} \) play vital roles. Given a Banach space X that includes an isomorphic copy of \(\backslash ell_1 \), we can use the dual relationship to find that X* will naturally contain an isomorphic copy of \(\backslash ell_{\backslash infty} \). By digging further into the properties of these spaces, we establish that X* contains \(\backslash ell_1(2^\mathbf{N}) \) which is more complex but follows from these underlying simpler sequence spaces.