Hahn-Banach and Dual Spaces - Core Definitions

The Hahn-Banach Theorem is one of the three pillars of functional analysis, alongside the Uniform Boundedness Principle and the Open Mapping Theorem. It guarantees the existence of continuous linear functionals with prescribed properties.

DefinitionSublinear Functional

Let $X$ be a vector space over $\mathbb{R}$ . A function $p : X \to \mathbb{R}$ is sublinear if:

Subadditivity: $p(x + y) \leq p(x) + p(y)$ for all $x, y \in X$
Positive Homogeneity: $p(\alpha x) = \alpha p(x)$ for all $x \in X$ and $\alpha \geq 0$

The norm on a normed space is an example of a sublinear functional.

TheoremHahn-Banach Theorem (Real Case)

Let $X$ be a real vector space, $p : X \to \mathbb{R}$ a sublinear functional, and $M \subset X$ a subspace. If $f : M \to \mathbb{R}$ is a linear functional satisfying $f(x) \leq p(x) \quad \text{for all } x \in M$ then there exists a linear extension $F : X \to \mathbb{R}$ of $f$ such that $F(x) \leq p(x) \quad \text{for all } x \in X$

The theorem states that any linear functional dominated by a sublinear functional on a subspace can be extended to the whole space while preserving the domination condition.

TheoremHahn-Banach Theorem (Normed Space Version)

Let $X$ be a normed space, $M \subset X$ a subspace, and $\phi : M \to \mathbb{K}$ a bounded linear functional. Then there exists $\Phi \in X^*$ such that:

$\Phi|_M = \phi$ (extension property)
$\|\Phi\| = \|\phi\|$ (norm preservation)

This version is perhaps the most useful in applications. It guarantees that bounded linear functionals on subspaces can be extended to the whole space without increasing the norm.

ExampleApplications of Hahn-Banach

Existence of Functionals: For any nonzero $x_0 \in X$ , there exists $\phi \in X^*$ with $\|\phi\| = 1$ and $\phi(x_0) = \|x_0\|$
Separation: If $x \neq y$ in a normed space, there exists $\phi \in X^*$ with $\phi(x) \neq \phi(y)$
Dual Space is Large: The dual space $X^*$ is large enough to separate points in $X$
Best Approximation: The distance from a point to a subspace can be computed using dual functionals

Remark

The Hahn-Banach Theorem is an existence result that relies on Zorn's Lemma (equivalent to the Axiom of Choice). It typically does not provide an explicit formula for the extension, but guarantees that such an extension exists.

This theorem is fundamental to the study of dual spaces, convex analysis, and optimization theory. It ensures that the dual space is "rich" enough to capture the geometry of the original space.