Question

Let \(X, Y\) be Banach spaces and \(T \in \mathcal{B}(X, Y)\). Show that: (i) \(T\) is an isomorphism into if and only if \(T^{* *}\) is an isomorphism into. (ii) \(T\) is an isometry into if and only if \(T^{* *}\) is an isometry into.

Short answer

T is an isomorphism into if and only if T** is an isomorphism into. T is an isometry into if and only if T** is an isometry into.

Step-by-step solution

Understand the Definitions

Recall that a map between Banach spaces is called an isomorphism into if it is a bijective bounded linear operator with a bounded inverse. An isometry into is a map that preserves the norm, i.e., \( \|T(x)\| = \|x\| \) for all \( x \) in the space.

Define the Double Dual

Understand that the double dual of a Banach space \( X \) is \( X^{**} \), and for a bounded linear operator \( T \), the double adjoint \( T^{**} \) is an operator acting on \( X^{**} \).

Prove (i) - T is an isomorphism into if and only if T** is an isomorphism into

First, assume \( T \) is an isomorphism into. This means \( T \) is injective, and there exists \( S \) such that \( ST = I \). Since \( S \) is bounded, \( S \) extends to \( S^{**} \), and \( S^{**}T^{**} = I \, \) thus \( T^{**} \) is also bounded and bijective, making it an isomorphism.

Prove (ii) - T is an isometry into if and only if T** is an isometry into

Assume \( T \) is an isometry into, which means it preserves the norm, i.e., \( \|T(x)\| = \|x\| \). It follows from the properties of Banach spaces that \( T^{**} \) also preserves the norm because dual spaces respect norm structures. Therefore, \( T^{**} \) is an isometry into.

Key concepts

isomorphism into

An isomorphism into is a special type of mapping between two Banach spaces—let's call them \(X\) and \(Y\). This map, denoted as \(T\), is not only linear but also bijective and bounded.
The bijective nature means that there exists a one-to-one correspondence between elements of \(X\) and elements of \(Y\). Essentially, there's no \(x\) in \(X\) or \(y\) in \(Y\) that gets left out.
For \(T\) to be bounded, a significant requirement is that there exists another operator, say \(S\), which acts as the inverse of \(T\). When you apply \(T\) followed by \(S\), you get back to where you started, much like how multiplying a number by its reciprocal gives 1. In mathematical terms, we write this as \(ST = I\), where \(I\) is the identity operator.
One important thing to note is that when proving the first part of the exercise—(i) \(T\) is an isomorphism into if and only if \(T^{**}\) is an isomorphism into— we effectively show that the original operator's isomorphism properties carry over to its double dual.

isometry into

An isometry into refers to a linear map \(T\) between two Banach spaces that preserves the norm. If you have a vector \(x\) in \(X\), applying an isometry \(T\) to \(x\) won't change its length. This property is mathematically captured by the equation \(\|T(x)\| = \|x\|\).
Why is this important? Norm-preserving maps maintain the geometric structure of the space. It means distances between points remain unchanged, making isometries very powerful in functional analysis.
In our exercise, proving (ii)—\(T\) is an isometry into if and only if \(T^{**}\) is an isometry into— rests on the foundation that double dual spaces retain the norm-preserving attributes of the original space. Essentially, this ensures that the norm in \(X\) matches precisely with the norm in its double dual \(X^{**}\), thereby maintaining the isometric property for the operator \(T\) as well as its double dual \(T^{**}\).

double dual

The double dual of a Banach space \(X\) is denoted as \(X^{**}\). To understand this, let's first consider the dual space \(X^*\), which consists of all continuous linear functionals mapping \(X\) into the base field (usually \(\textbf{R}\) or \(\textbf{C}\)). The double dual \(X^{**}\) then, is the dual space of the dual space \(X^*\).
So, what does that mean in simpler terms? Essentially, \(X^{**}\) contains functionals that act on elements of \(X^*\).
The beauty of the double dual lies in the fact that there always exists a natural embedding from \(X\) into \(X^{**}\). This embedding allows us to treat \(X\) as a subspace within \(X^{**}\), highlighting deep and rich structural nuances of Banach spaces.
In the context of the given exercise, we explored that for a bounded linear operator \(T\) on a Banach space, its double adjoint \(T^{**}\) also operates in the same manner on \(X^{**}\). This underlying principle forms the crux of our proofs, showing that isomorphism and isometry properties of \(T\) are mirrored in \(T^{**}\).