The Topological Dual of a Normed Linear Space

# The Topological Dual of a Normed Linear Space

Recall that if $X$ is a linear space then we defined the algebraic dual of $X$ denoted by $X^{\#}$ to be the linear space of all linear functionals on $X$.

Now suppose that $X$ is a normed linear space. Then we can talk about continuity. As we saw on The Normed Linear Space B(X, Y) page, we can talk about the set of all continuous linear functionals, that is, the set of all continuous linear operators from $X$ to $\mathbb{C}$ which is the normed linear space $\mathcal B(X, \mathbb{C})$. We give this space a special name.

 Definition: Let $X$ be a normed linear space. Then the normed linear space of all continuous linear functionals on $X$ is the Topological Dual of $X$ and is denoted by $X^*$.

Observe that the topological dual $X^*$ is a subspace of the algebraic dual $X^{\#}$.

 Theorem 1: Let $X$ be a normed linear space. Then the topological dual $X^*$ is a Banach space.
• Proof: Clearly $\mathbb{C}$ is a Banach space and by definition:
(1)
\begin{align} \quad X^* = \mathcal B(X, \mathbb{C}) \end{align}