2.2 Linear functionals and dual spaces

Linear functionals are key players in functional analysis, mapping elements from normed spaces to their scalar fields. They're like special measuring tools, giving us a single number for each element in a space.

The dual space, made up of all bounded linear functionals, is a powerful concept. It helps us understand the original space better, often revealing hidden structures and properties we might have missed otherwise.

Linear Functionals

Linear functionals on normed spaces

• Linear functionals map elements from a normed space $X$ to its underlying scalar field $\mathbb{F}$ ($\mathbb{R}$ or $\mathbb{C}$)
• Satisfy additivity property: $f(x + y) = f(x) + f(y)$ for all $x, y \in X$
• Satisfy homogeneity property: $f(\alpha x) = \alpha f(x)$ for all $x \in X$ and $\alpha \in \mathbb{F}$
• Example: On $C[a, b]$, the evaluation functional $f_t(x) = x(t)$ for fixed $t \in [a, b]$
• Example: On $\ell^p$, the coordinate functional $f_n(x) = x_n$ for fixed $n \in \mathbb{N}$
• Example: On $\mathcal{P}$, the derivative functional $f(p) = p'(0)$

Dual space of normed spaces

• Dual space of a normed space $X$, denoted $X^*$, consists of all bounded linear functionals on $X$
• $X^*$ is a normed space with the norm: $\|f\| = \sup\{|f(x)| : x \in X, \|x\| \leq 1\}$
• $X^*$ is a Banach space (complete normed space) if $X$ is a Banach space
• If $X$ is a Hilbert space, $X^*$ is isometrically isomorphic to $X$
• Dual space of a finite-dimensional normed space is isomorphic to the space itself

Boundedness and norm of functionals

• A linear functional $f$ on a normed space $X$ is bounded if there exists $M \geq 0$ such that: $|f(x)| \leq M\|x\|$ for all $x \in X$
• Norm of a bounded linear functional $f$ is defined as: $\|f\| = \inf\{M \geq 0 : |f(x)| \leq M\|x\|$ for all $x \in X\}$
• Equivalent definition: $\|f\| = \sup\{|f(x)| : x \in X, \|x\| \leq 1\}$
• To find the norm of a linear functional:
  1. Find an upper bound $M$ for $|f(x)|$ in terms of $\|x\|$
  2. Prove the upper bound is sharp by finding a sequence $(x_n)$ in $X$ with $\|x_n\| = 1$ such that $|f(x_n)| \to M$ as $n \to \infty$

Normed spaces vs dual spaces

• Canonical isomorphism $J: X \to X^{**}$ (dual of the dual space) defined by: $J(x)(f) = f(x)$ for all $x \in X$ and $f \in X^*$
• $J$ is a linear isometry: $\|J(x)\| = \|x\|$ for all $x \in X$
• $J$ is injective: if $J(x) = J(y)$, then $x = y$
• For a Banach space $X$, $J$ is an isometric isomorphism if and only if $X$ is reflexive ($J$ is surjective)
• Canonical isomorphism allows studying the relationship between a normed space and its dual
• Plays a crucial role in the theory of reflexive spaces