🧐Functional Analysis Unit 9 – Weak and Weak* Topologies

Key Concepts and Definitions

• Topology defines the notion of open sets, closed sets, and continuity in a space
• Weak topology is the coarsest topology that makes a family of linear functionals continuous
• Weak* topology is the weak topology on the dual space of a normed linear space
• Locally convex space is a vector space with a topology defined by a family of seminorms
• Separable space has a countable dense subset
• Reflexive space is isomorphic to its double dual under the natural map
• Banach space is a complete normed linear space

Weak Topology Basics

• Weak topology is generated by the family of seminorms $p_f(x) = |f(x)|$, where $f$ belongs to the dual space
• A set is weakly open if it is a union of finite intersections of sets of the form $\{x: |f(x) - f(x_0)| < \epsilon\}$
• Weakly continuous functions are continuous with respect to the weak topology
• Weak convergence of a sequence $x_n$ to $x$ means $f(x_n) \to f(x)$ for all $f$ in the dual space
  • Denoted as $x_n \rightharpoonup x$
• Weak topology is always coarser than the norm topology
  • Every weakly open set is norm open, but not conversely
• Weak topology coincides with the norm topology on finite-dimensional spaces

Weak* Topology Fundamentals

• Weak* topology is the topology of pointwise convergence on the dual space
• A subbase for the weak* topology consists of sets of the form $\{f: |f(x) - f_0(x)| < \epsilon\}$ for fixed $x$ in the original space
• Weak* convergence of a sequence $f_n$ to $f$ means $f_n(x) \to f(x)$ for all $x$ in the original space
  • Denoted as $f_n \overset{*}{\rightharpoonup} f$
• Weak* topology is always coarser than the norm topology on the dual space
• Weak* topology coincides with the weak topology on reflexive spaces
• Banach-Alaoglu theorem states that the unit ball of the dual space is weak* compact

Comparison of Weak and Weak* Topologies

• Weak topology is defined on a normed linear space, while weak* topology is defined on its dual space
• Weak convergence is tested against all functionals in the dual space, while weak* convergence is tested against all elements in the original space
• Weak topology is generally stronger than the weak* topology when both are defined
  • Every weak* open set is weakly open, but not conversely
• In reflexive spaces, the weak and weak* topologies coincide
• Weak topology is more relevant for studying properties of the original space, while weak* topology is more relevant for studying properties of the dual space

Properties and Theorems

• Banach-Alaoglu theorem weak* compactness of the unit ball in the dual space
• Goldstine's theorem unit ball of a Banach space is weakly dense in the unit ball of its double dual
• Eberlein-Šmulian theorem characterizes weak compactness in terms of weak sequential compactness
• Krein-Milman theorem every compact convex set is the closed convex hull of its extreme points
  • Particularly useful in the weak* topology
• Banach-Steinhaus theorem (uniform boundedness principle) a pointwise bounded family of operators is uniformly bounded
• Mazur's lemma convex closure of a weakly compact set is weakly compact

Applications in Functional Analysis

• Weak topology is used to study convergence and compactness properties of sets and sequences
  • Useful in proving existence of solutions to variational problems
• Weak* topology is used to study properties of the dual space and its relation to the original space
  • Useful in representing linear functionals and studying their continuity
• Weak and weak* topologies play a crucial role in the theory of Banach spaces and operator theory
  • Used in the study of spectral theory, ergodic theory, and harmonic analysis
• Weak convergence is important in the study of partial differential equations and optimization
  • Provides a notion of convergence that is often more suitable than norm convergence

Examples and Problem-Solving Strategies

• To show a set is weakly open, express it as a union of finite intersections of sets of the form $\{x: |f(x) - f(x_0)| < \epsilon\}$
• To show a sequence converges weakly, verify that $f(x_n) \to f(x)$ for all $f$ in the dual space
  • Often easier than proving norm convergence
• To show a set is weakly compact, use the Eberlein-Šmulian theorem and prove weak sequential compactness
• To find extreme points of a weakly compact set, use the Krein-Milman theorem
• When working with the dual space, consider using the weak* topology and its properties
  • Banach-Alaoglu theorem is particularly useful for proving weak* compactness

Advanced Topics and Extensions

• Mackey topology is the strongest locally convex topology that agrees with the weak topology on bounded sets
• Arens-Mackey theorem characterizes the Mackey topology in terms of convergence of bounded nets
• Gelfand-Naimark-Segal (GNS) construction builds a Hilbert space from a C*-algebra using the weak* topology
• Weak and weak* topologies can be generalized to locally convex spaces using the family of seminorms
• Baire category theorem has important implications for the study of weak and weak* topologies
  • Used to prove the Banach-Steinhaus theorem and other results
• Choquet theory studies the representation of points in compact convex sets using probability measures
  • Closely related to the Krein-Milman theorem and the weak* topology