Hahn-Banach Theorem

• categories: Functional analysis, Theorem

The Hahn-Banach Theorem is a fundamental result in functional analysis. It allows the extension of a boundedLinear Functional defined on a subspace of a vector space to the whole space, without increasing its norm.

Statement

Let $X$ be a normed vector space over $\mathbb{R}$ or $\mathbb{C}$, and let $U\subset X$ be a subspace. If $f_0:U\to \mathbb{R}$ is a bounded linear functional with norm $\Vert f_0\Vert \le C$, then there exists an extension $f:X\to \mathbb{R}$ with $\Vert f\Vert =\Vert f_0\Vert$.

Key Properties and Implications

1. Extension of Linear Functionals: Any linear functional defined on a subspace can be extended to the whole space without increasing its norm.
2. Separation of Convex Sets: The Hahn-Banach Theorem implies that disjoint convex sets can often be separated by a hyperplane, an important tool in optimization and convex analysis.
3. Dual Space Density: In a normed space $X$, the Dual Space $X^{\ast }$ is rich in functionals, making it possible to approximate points and study Compactness.

Intuition

The theorem guarantees that “small” information (a linear functional on a subspace) can be extended consistently to “larger” contexts (the entire space) without losing control over its behavior, making it crucial for defining and studying dual spaces and continuous functionals.

Outline of Proof (Normed Space Version)

1. Extension Process: Begin with $f_0$ on $U$. Gradually extend $f_0$ to larger subspaces by defining values on vectors outside $U$ while maintaining boundedness.
2. Zorn’s Lemma: The construction usesZorn’s Lemma to ensure the extension can be made over all of $X$ while keeping $\Vert f\Vert =\Vert f_0\Vert$.