The Dual-Space

There are two methods to studying mathematical structures.

1. By looking at the objects in the space.

2. By looking at functions on the space.

Let’s see how the second setting looks like in our Banach space setting.

Introduction

In the Banach space setting the natural mappings to consider are the linear continuous functions between two Banach spaces \(X\) and \(Y\). Now consider \(\mathcal{L}(X, \mathbb{R})\) the space of bounded linear continuous functionals on \(X\). This is a Banach space, and we will call it the dual of \(X\) denotes by \(X^*\).

The intuitive view is that functionals are measurements of an object in the space \(X\).

Example 12

We encounter many functionals in daily life.

1. A scale measures the weight of objects i.e. it assigns a real number to each object.

2. A rules measures length.

3. CT-scanners measure line integrals of an unknown tissue density.

4. MRI-scanners measure proton densities along lines.

Isomorphisms and Isometries

First let’s explore how we show that two Banach spaces have the same structure. In normed vectorspaces the natual map is \(T : X \mapsto Y\). \(X, Y\) are equivalent if there exists an invertible continuous linear map \(T : X \mapsto Y\) i.e.

\[ c \norm{x}_X \leq \norm{Tx}_Y \leq C \norm{x}_X \]

Is such a map \(T\) exists we say that \(X\) and \(Y\) are isomorphic. If \(c = C = 1\) \(T\) is an an isometry.

Intuition Building in \(\mathbb{R}^2\).