3.3 Consequences for linear functionals and dual spaces

Dual spaces and the Hahn-Banach Theorem are key concepts in functional analysis. They provide powerful tools for understanding normed linear spaces through their linear functionals, extending bounded linear maps, and separating points and sets.

These ideas have far-reaching implications in analysis and optimization. They form the foundation for weak topologies, reflexivity of spaces, and the study of convex sets, which are crucial in many areas of mathematics and its applications.

Dual Spaces and the Hahn-Banach Theorem

Existence of bounded linear functionals

• A normed linear space $(X, \|\cdot\|)$ consists of a vector space $X$ equipped with a norm $\|\cdot\|$ that measures the "size" of elements in $X$
  • The norm satisfies properties such as positive definiteness ($\|x\| = 0$ iff $x = 0$), scalar multiplication compatibility ($\|\alpha x\| = |\alpha| \|x\|$), and the triangle inequality ($\|x + y\| \leq \|x\| + \|y\|$)
• Linear functionals $f: X \to \mathbb{R}$ (or $\mathbb{C}$) are linear maps that preserve vector space operations ($f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)$)
• Bounded linear functionals have their output controlled by the norm of the input ($|f(x)| \leq M \|x\|$ for some constant $M \geq 0$)
• The Hahn-Banach Theorem guarantees the existence of non-zero bounded linear functionals on any normed linear space ($\ell^p$ spaces, $L^p$ spaces)

Dual space of normed spaces

• The dual space $X^*$ of a normed linear space $(X, \|\cdot\|)$ contains all bounded linear functionals on $X$
  • Dual spaces provide a way to study properties of the original space $X$ through its linear functionals
• $X^*$ forms a normed linear space with the operator norm $\|f\|_{X^*} = \sup\{|f(x)| : \|x\| \leq 1\}$
• If $X$ is a Banach space (complete normed linear space), then $X^*$ is also a Banach space
• The dual space separates points in $X$: for distinct $x, y \in X$, there exists $f \in X^*$ with $f(x) \neq f(y)$ ($c_0$ and $\ell^1$)

Hahn-Banach Theorem for dual spaces

• The Hahn-Banach Theorem allows extending bounded linear functionals from a subspace to the whole space
  • If $Y$ is a subspace of a normed linear space $X$ and $f: Y \to \mathbb{R}$ (or $\mathbb{C}$) is bounded, then $f$ extends to a bounded linear functional $F: X \to \mathbb{R}$ (or $\mathbb{C}$) with the same norm
• Proof outline:
  1. Use Zorn's Lemma to obtain a maximal extension of $f$
  2. Show that the maximal extension has the same norm as $f$
  3. Prove that the maximal extension is defined on the entire space $X$
• The theorem has significant implications in functional analysis and optimization ($L^p$ spaces, $C[a,b]$)

Normed spaces vs second duals

• The second dual space $X^{**}$ is the dual of the dual space $X^*$
• The canonical embedding $J: X \to X^{**}$ is defined by $J(x)(f) = f(x)$ for $x \in X$ and $f \in X^*$
  • $J$ preserves the norm ($\|J(x)\|_{X^{**}} = \|x\|_X$) and is injective (one-to-one)
• A normed space $X$ is reflexive if $J$ is surjective (onto), i.e., $X$ is isometrically isomorphic to $X^{**}$
  • Reflexive spaces include Hilbert spaces and $L^p$ spaces for $1 < p < \infty$
  • Non-reflexive spaces include $L^1$, $L^\infty$, and $c_0$

Implications for weak topologies

• The weak topology on $X$ is the coarsest topology making all functionals in $X^*$ continuous
  • A net $(x_\alpha)$ converges weakly to $x$ in $X$ iff $f(x_\alpha) \to f(x)$ for all $f \in X^*$
• The weak* topology on $X^*$ is the coarsest topology making all evaluations $f \mapsto f(x)$ continuous for each $x \in X$
  • A net $(f_\alpha)$ converges weak* to $f$ in $X^*$ iff $f_\alpha(x) \to f(x)$ for all $x \in X$
• The Banach-Alaoglu Theorem states that the closed unit ball of $X^*$ is compact in the weak* topology
• The Hahn-Banach Theorem enables the separation of convex sets in locally convex spaces ($L^p$ spaces, Hilbert spaces)
  • Locally convex spaces are characterized by a family of seminorms and include all normed spaces