More on Dual Spaces

Reminder and Limits

In one of the previous chapters, we defined the dual space of a normed space $X$ as the set of all bounded linear transformations $T : X \to \mathbb{R},$ equipped with the norm $\lVert T\rVert = \sup_{\|x\| \le 1} |T(x)|.$ This space, denoted by $X'$ or $B(X,\mathbb{R})$, is itself a normed space.

However, this framework has its limitations. Not every space of interest in analysis can be endowed with a norm that captures the desired notion of convergence. In such cases, the concept of boundedness is no longer adequate, and continuity must be interpreted in a broader, topological sense.

This leads naturally to the idea of topological vector spaces and their corresponding topological duals.

An Overview of Topological Vector Spaces

Let $X$ be a set. A topology on $X$ is a collection $\tau$ of subsets of $X$ (called open sets) satisfying the following properties:

1. $\varnothing, X \in \tau$.
2. The union of any collection of sets in $\tau$ belongs to $\tau$.
3. The intersection of any finite collection of sets in $\tau$ belongs to $\tau$.

The pair $(X, \tau)$ is called a topological space.

A topological space $(X, \tau)$ is called Hausdorff space if every pair of distinct points $x, y \in X$ have disjoint neighborhoods.

Topological Vector Spaces

A topological vector space (TVS) over $\mathbb{R}$ is a Hausdorff space that is also a vector space $X$ such that for all $x,y\in X$ and $\alpha\in\mathbb{R}$:

1. The vector addition $(x, y) \mapsto x + y$ is continuous with respect to the product topology on $X \times X$.

2. The scalar multiplication $(\alpha, x) \mapsto \alpha x$ is continuous with respect to the product topology on $\mathbb{R} \times X$.

Continuous Linear Functional and Topological Dual

Let $X$ be a topological vector space over $\mathbb{R}$. A linear functional on $X$ is a linear map $T : X \to \mathbb{R}.$

The functional $T$ is said to be continuous if it is continuous with respect to the topology of $X$ and the usual topology on $\mathbb{R}$. Equivalently, $T$ is continuous if for every sequence $\{x_n\}$ in $X$ that converges to $x$, we have

$$x_n \to x \quad \Rightarrow \quad T(x_n) \to T(x).$$

Topological Dual

Let $X$ be a topological vector space over $\mathbb{R}.$ The topological dual of $X$, denoted by $X'$, is the set of all continuous linear functionals on $X$, that is,

$$X' = \{\, T : X \to \mathbb{R} \mid T \text{ is linear and continuous} \,\}.$$

Each element of $X'$ is called a continuous linear functional on $X$.

Example

If $X$ is a normed space, continuity is equivalent to boundedness, and therefore $X' = B(X, \mathbb{R})$. Where $B(X, \mathbb{R})$ denotes the space of all bounded linear transformation from $X$ into $\mathbb{R}$.