Hahn-Banach and Dual Spaces - Key Properties

The dual space of a Banach space carries a rich structure that reflects and generalizes properties of the original space. Understanding this relationship is central to functional analysis.

DefinitionDual Space

Let $X$ be a normed space. The dual space $X^*$ is the set of all bounded (continuous) linear functionals $\phi : X \to \mathbb{K}$ .

With the operator norm $\|\phi\| = \sup_{\|x\| \leq 1} |\phi(x)|$ , the space $X^*$ is a Banach space, even if $X$ is not complete.

DefinitionReflexive Spaces

The bidual or second dual of $X$ is $X^{**} = (X^*)^*$ . There is a natural embedding $J : X \to X^{**}$ defined by $(J(x))(\phi) = \phi(x)$ for all $\phi \in X^*$ . The map $J$ is an isometric linear embedding: $\|J(x)\| = \|x\|$ .

A Banach space $X$ is reflexive if $J : X \to X^{**}$ is surjective.

ExampleDual Spaces of Standard Spaces

Finite Dimensions: If $X = \mathbb{K}^n$ , then $X^* \cong \mathbb{K}^n$ and $X$ is reflexive
$\ell^p$ Spaces: For $1 < p < \infty$ , we have $(\ell^p)^* \cong \ell^q$ where $1/p + 1/q = 1$ . These spaces are reflexive
$\ell^1$ and $\ell^\infty$ : $(\ell^1)^* \cong \ell^\infty$ but $(\ell^\infty)^* \supsetneq \ell^1$ . Neither $\ell^1$ nor $\ell^\infty$ is reflexive
$L^p$ Spaces: For $1 < p < \infty$ , $(L^p)^* \cong L^q$ with $1/p + 1/q = 1$ , and $L^p$ is reflexive
$C[0,1]$ : The dual is $M[0,1]$ , the space of signed Borel measures, by the Riesz Representation Theorem. $C[0,1]$ is not reflexive

TheoremProperties of Reflexive Spaces

Let $X$ be a Banach space. Then:

If $X$ is reflexive, then $X$ is complete
If $X$ is reflexive, then $X^*$ is reflexive
Hilbert spaces are reflexive
Finite-dimensional spaces are reflexive
A closed subspace of a reflexive space is reflexive

Remark

Reflexivity is a strong structural property. Reflexive spaces have particularly nice compactness and weak convergence properties. For instance, in a reflexive space, every bounded sequence has a weakly convergent subsequence.

ExampleSeparation of Convex Sets

Let $C$ and $D$ be disjoint closed convex sets in a normed space $X$ with $C$ compact. Then there exists $\phi \in X^*$ and $\alpha \in \mathbb{R}$ such that $\phi(x) < \alpha < \phi(y)$ for all $x \in C$ and $y \in D$ .

This geometric separation result follows from the Hahn-Banach Theorem and is fundamental to convex optimization.

The study of dual spaces and reflexivity reveals deep connections between geometry, topology, and analysis.