Question

Let \(X, Y\) be Banach spaces, \(T \in \mathcal{B}(X, Y)\). Show that if \(T\) is an isomorphism of \(X\) onto \(Y\), then \(T^{* *}(X)=Y\) and \(\left(T^{* *}\right)^{-1}(Y)=X\).

Short answer

If T is an isomorphism of X onto Y, then T**(X) = Y and (T**)^{-1}(Y) = X.

Step-by-step solution

- Understand the Problem

Given two Banach spaces, X and Y, and an operator T that is an isomorphism of X onto Y, we need to show two things: (1) T**(X) = Y, and (2) (T**)^{-1}(Y) = X. An operator T being an isomorphism means T is a bounded, bijective linear map with a bounded inverse.

- Consider Dual Spaces

For Banach spaces, the dual space of X, denoted X*, consists of all continuous linear functionals on X. The bidual space, denoted X**, is the dual of X*.

- Define T* and T**

The adjoint operator T*, mapping Y* to X*, is defined by \(T^*(y^*)(x) = y^*(Tx)\) for all x in X and y* in Y*. The double adjoint operator T** maps X** to Y**.

- Use Isomorphism Properties

Since T is an isomorphism, T* is also an isomorphism from Y* onto X*. This implies that both T and T* have bounded inverses, noted as T^{-1} and (T*)^{-1} respectively.

- Connect T and T**

For any \(x^{**} \in X^{**}\), define \(T^{**}(x^{**}) \in Y^{**}\) by \(T^{**}(x^{**})(y^*) = x^{**}(T^*(y^*))\). Since T is an isomorphism, T** is also bijective.

- Demonstrate T**(X) = Y

Since X can be isomorphically embedded into its bidual X** and T is bijective, for every y ∈ Y, there exists an x in X such that T(x) = y. Thus, T**(X) = Y by definition of T**.

- Show (T**)^{-1}(Y) = X

From step 6 and the property of isomorphisms, the inverse of T**, which maps elements of Y back to X, will satisfy (T**)^{-1}(Y) = X, because T maps surjectively and bijectively.

Key concepts

Banach spaces

A Banach space is a complete normed vector space. This means it is a vector space equipped with a norm, and every Cauchy sequence in this space converges to an element within the space. Some key points about Banach spaces are:

• They are fundamental in functional analysis and related theories.
• Completeness is crucial as it ensures that limit points are well-defined within the same space.
• Examples include spaces like \( \ell^p \) spaces and the space of continuous functions on a closed interval.

Understanding Banach spaces is essential when dealing with operators and their properties, as is evident in the exercise where \( X \) and \( Y \) are Banach spaces.

Isomorphism

In the context of Banach spaces, an isomorphism is a bounded, bijective linear map with a bounded inverse. This implies a few important characteristics:

• The operator preserves the structure of the spaces, meaning linearity and boundedness are maintained.
• Bijection ensures a one-to-one correspondence between elements of \( X \) and \( Y \).
• Having a bounded inverse ensures stability in the process of mapping back and forth between the spaces.

In our problem, the operator \( T \) is such an isomorphism. This enables the transfer of properties between \( X \) and \( Y \) effectively.

Adjoint operator

The adjoint operator \( T^* \) is a key concept when dealing with dual and bidual spaces. It maps from the dual space \( Y^* \) to \( X^* \) and is defined by the relation: \( T^*(y^*)(x) = y^*(T(x)) \) for all \( x \in X \) and \( y^* \in Y^* \). Important aspects of \( T^* \) are:

• It preserves linearity and continuity, essential for analyzing properties of the underlying spaces.
• In this exercise, since \( T \) is an isomorphism, \( T^* \) also holds similar bijective and bounded nature.
• The relationship between \( T \) and \( T^* \) allows us to extend the isomorphism property to the dual spaces

Bidual space

The bidual space of a Banach space \( X \), denoted \( X^{**} \), is the dual of \( X^* \). It means \( X^{**} \) consists of all continuous linear functionals on \( X^* \). Here are some key points about bidual spaces:

• They provide a natural extension of a Banach space.
• Every Banach space \( X \) can be isomorphically embedded into its bidual \( X^{**} \).
• The exercise showed that the double adjoint operator \( T^{**} \) maps \( X^{**} \) to \( Y^{**} \).

By analyzing bidual spaces, we gain deeper insights into the underlying structures of Banach spaces and their duals.

Dual space

The dual space of a Banach space \( X \), denoted \( X^* \), consists of all continuous linear functionals on \( X \). Understanding the dual space is critical in functional analysis:

• It captures essential properties of \( X \) through the lenses of linear functionals.
• The adjoint operator \( T^* \) operates between the dual spaces of \( Y \) and \( X \), maintaining the isomorphism property.
• In the context of the exercise, knowing \( X^* \) and \( Y^* \) allowed us to define and utilize \( T^* \) and \( T^{**} \).

Thus, understanding the dual space lays the groundwork for more complex mappings and extensions like the adjoint and double adjoint operators.