13.1 Convex analysis in Banach spaces

Convex sets and functions play a crucial role in Banach spaces. They're the building blocks for many important theorems and applications in functional analysis. Understanding these concepts is key to grasping more advanced topics.

The Hahn-Banach theorem is a powerful tool for extending linear functionals and separating convex sets. It has far-reaching consequences, including the existence of non-trivial dual spaces and the characterization of reflexive Banach spaces.

Convex Sets and Functions in Banach Spaces

Convex sets in Banach spaces

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• Convex set contains all points on the line segment connecting any two points within the set
• Examples of convex sets in Banach spaces include closed balls $B[x_0, r] = \{x \in X: \|x - x_0\| \leq r\}$, hyperplanes $\{x \in X: f(x) = \alpha\}$ where $f$ is a continuous linear functional and $\alpha \in \mathbb{R}$, and halfspaces $\{x \in X: f(x) \leq \alpha\}$ or $\{x \in X: f(x) \geq \alpha\}$

Convex functions in Banach spaces

• Convex function satisfies the inequality $f((1-t)x + ty) \leq (1-t)f(x) + tf(y)$ for any $x, y \in X$ and $t \in [0, 1]$, meaning the line segment connecting any two points on the graph lies above or on the graph
• Examples of convex functions in Banach spaces include norms $f(x) = \|x\|$, continuous linear functionals $f(x) = \langle x^*, x \rangle$ where $x^* \in X^*$ (dual space), and affine functions $f(x) = \langle x^*, x \rangle + \alpha$ where $x^* \in X^*$ and $\alpha \in \mathbb{R}$

Hahn-Banach Theorem and Its Consequences

Hahn-Banach theorem applications

• Hahn-Banach theorem (analytic form) extends any continuous linear functional defined on a subspace to the whole space without increasing its norm
• Hahn-Banach theorem (geometric form) separates any closed convex set and a point outside it by a hyperplane
• Consequences of the Hahn-Banach theorem:
  1. Normed spaces have enough continuous linear functionals to separate points
  2. The dual space $X^*$ of a Banach space $X$ is non-trivial ($X^* \neq \{0\}$)
  3. The bidual space $X^{**}$ is isometrically isomorphic to $X$ for reflexive Banach spaces

Subdifferentials in Banach Spaces

Subdifferentials in Banach spaces

• Subdifferential of a convex function $f$ at $x \in X$ is the set $\partial f(x) = \{x^* \in X^*: f(y) \geq f(x) + \langle x^*, y-x \rangle, \forall y \in X\}$, generalizing the concept of the gradient for non-differentiable convex functions
• Properties of subdifferentials:
  • If $f$ is differentiable at $x$, then $\partial f(x) = \{\nabla f(x)\}$
  • If $f$ is continuous at $x$, then $\partial f(x)$ is a non-empty, convex, and weak* compact subset of $X^*$
  • The subdifferential is a monotone operator, satisfying $\langle x^* - y^*, x - y \rangle \geq 0$ for any $x, y \in X$, $x^* \in \partial f(x)$, and $y^* \in \partial f(y)$
• Applications of subdifferentials include optimality conditions ($x$ is a minimizer of $f$ if and only if $0 \in \partial f(x)$) and characterizing proximal operators $\operatorname{prox}_f(x) = \operatorname{argmin}_{y \in X} \{f(y) + \frac{1}{2}\|y-x\|^2\}$

Convex Conjugate Functions

Properties of convex conjugates

• Convex conjugate (or Fenchel conjugate) of a convex function $f$ is defined as $f^*(x^*) = \sup_{x \in X} \{\langle x^*, x \rangle - f(x)\}$ and is always convex, even if the original function is not
• Fenchel-Young inequality: $f(x) + f^*(x^*) \geq \langle x^*, x \rangle$ for all $x \in X$ and $x^* \in X^*$
• Double conjugate: $(f^*)^* = f$ if and only if $f$ is convex, lower semicontinuous, and proper
• Subdifferential characterization: $x^* \in \partial f(x)$ if and only if $f(x) + f^*(x^*) = \langle x^*, x \rangle$

Examples of convex conjugates

• Indicator function $\delta_C(x) = \begin{cases} 0, & x \in C \\ +\infty, & x \notin C \end{cases}$ where $C$ is a closed convex set has conjugate equal to the support function $\delta_C^*(x^*) = \sup_{x \in C} \langle x^*, x \rangle$
• Norm $f(x) = \|x\|$ has conjugate equal to the indicator function of the unit ball in the dual space, $f^*(x^*) = \delta_{B[0, 1]}(x^*)$
• Quadratic function $f(x) = \frac{1}{2}\|x\|^2$ has conjugate that is also a quadratic function, $f^*(x^*) = \frac{1}{2}\|x^*\|^2$