10.3 Duality mappings and their applications

Duality mappings connect Banach spaces to their dual spaces, generalizing gradients from Hilbert spaces. They link each point to supporting functionals of the unit ball, offering insights into geometric properties and smoothness of Banach spaces.

These mappings play a crucial role in fixed point theory and nonlinear analysis. They help characterize uniform convexity and smoothness, and are key in studying nonexpansive mappings and accretive operators in Banach spaces.

Duality Mappings in Banach Spaces

Definition of duality mapping

• Maps a Banach space $X$ to its dual space $X^*$ associates each point $x \in X$ with the set of supporting functionals of the unit ball at $\frac{x}{\|x\|}$
  • Supporting functionals linear functionals that attain their maximum value on the unit ball at a given point (e.g., the hyperplanes tangent to the unit ball)
• Formally defined as $J(x) = \{x^* \in X^* : \langle x^*, x \rangle = \|x\|^2 = \|x^*\|^2\}$ where $\langle \cdot, \cdot \rangle$ denotes the duality pairing between $X$ and $X^*$
• Generalizes the concept of the gradient of the norm in Hilbert spaces to Banach spaces
  • In Hilbert spaces, the duality mapping coincides with the Riesz representation (e.g., $J(x) = x$ for all $x \in X$)

Existence in smooth Banach spaces

• Banach space $X$ is smooth if its norm is Gâteaux differentiable at every non-zero point
  • Gâteaux differentiability weaker form of differentiability than Fréchet differentiability
• In smooth Banach spaces, the duality mapping $J$ is single-valued and norm-to-weak* continuous
  • Single-valued for each $x \in X$, $J(x)$ contains exactly one element (e.g., $J(x) = \{x^*\}$ for some unique $x^* \in X^*$)
  • Norm-to-weak* continuous if $x_n \rightarrow x$ in norm, then $J(x_n) \rightharpoonup^* J(x)$ (weak* convergence in $X^*$)
• Existence and uniqueness follow from the Hahn-Banach theorem and the Gâteaux differentiability of the norm
  • Hahn-Banach theorem guarantees the existence of supporting functionals
  • Gâteaux differentiability ensures the uniqueness of the supporting functional at each point

Applications of Duality Mappings

Applications to Banach space geometry

• Characterizes geometric properties of Banach spaces such as uniform convexity and uniform smoothness
  • Uniform convexity $X$ is uniformly convex iff $J$ is uniformly monotone, i.e., $\langle J(x) - J(y), x - y \rangle \geq \delta(\|x - y\|)$ for some $\delta:(0,\infty) \rightarrow (0,\infty)$
  • Uniform smoothness $X$ is uniformly smooth iff $J$ is uniformly continuous on bounded sets
• Modulus of convexity and modulus of smoothness can be expressed in terms of the duality mapping
  • Moduli quantify the degree of uniform convexity and uniform smoothness ($L^p$ spaces for $1 < p < \infty$ are uniformly convex and uniformly smooth)

Use in fixed point theory

• Crucial tool in studying fixed point theorems for nonlinear mappings in Banach spaces
  • Fixed point a point $x \in X$ such that $T(x) = x$ for a mapping $T:X \rightarrow X$
• Nonexpansive mappings mappings $T:X \rightarrow X$ satisfying $\|T(x) - T(y)\| \leq \|x - y\|$ for all $x,y \in X$
  • In uniformly convex Banach spaces, every nonexpansive mapping with a bounded orbit has a fixed point (proof uses duality mapping and modulus of convexity)
• Accretive operators operators $A:X \rightarrow 2^X$ satisfying $\langle J(x - y), u - v \rangle \geq 0$ for all $x,y \in X$, $u \in A(x)$, and $v \in A(y)$
  • Zeros of accretive operators (points $x \in X$ such that $0 \in A(x)$) studied using duality mapping and monotone operator theory