🧐Functional Analysis Unit 10 – Duality Theory and Reflexive Spaces

Key Concepts and Definitions

• Dual space $X^*$ consists of all bounded linear functionals on a normed space $X$
• Linear functional $f: X \to \mathbb{R}$ or $\mathbb{C}$ preserves vector space structure ($f(ax+by) = af(x)+bf(y)$)
• Norm of a linear functional $\|f\| = \sup\{|f(x)|: \|x\| \leq 1\}$ measures its "size" or "magnitude"
• Weak topology on $X$ generated by the family of seminorms $\{p_f(x) = |f(x)|: f \in X^*\}$
  • Weakest topology making all elements of $X^*$ continuous
• Weak* topology on $X^*$ generated by the family of seminorms $\{p_x(f) = |f(x)|: x \in X\}$
  • Weakest topology making all evaluation functionals $e_x(f) = f(x)$ continuous
• Reflexive space $X$ isomorphic to its double dual $X^{**}$ via the canonical map $J: X \to X^{**}$

Dual Spaces and Linear Functionals

• Dual space $X^*$ forms a normed vector space with the operator norm $\|f\| = \sup\{\|f(x)\|: \|x\| \leq 1\}$
• Bounded linear functionals capture the notion of continuous linear maps from $X$ to its scalar field
• Evaluation of a linear functional $f$ at a point $x \in X$ given by $f(x)$, a scalar value
• Hahn-Banach theorem guarantees existence of non-trivial bounded linear functionals on any normed space
  • Allows extension of bounded linear functionals from subspaces to the whole space
• Riesz representation theorem identifies the dual of certain function spaces ($L^p, C[a,b]$) with other function spaces
• Dual of the dual space $X^{**}$ called the double dual or bidual of $X$

Normed Spaces and Their Duals

• Normed space $(X, \|\cdot\|)$ a vector space $X$ equipped with a norm $\|\cdot\|$ satisfying positivity, homogeneity, and triangle inequality
• Dual norm on $X^*$ defined by $\|f\|_{X^*} = \sup\{|f(x)|: \|x\|_X \leq 1\}$
  • Makes $X^*$ a normed space, allowing study of its topological and geometric properties
• Banach space a complete normed space (Cauchy sequences converge)
  • Dual of a Banach space is always a Banach space
• Separable normed space contains a countable dense subset ($\ell^p, L^p, C[a,b]$)
  • Separability not always inherited by the dual space ($\ell^1$ separable but $\ell^\infty$ not)
• Reflexive spaces ($\ell^p$ for $1 < p < \infty$) have duals that are "compatible" with the original space
  • Non-reflexive spaces ($\ell^1, c_0, L^1, C[a,b]$) exhibit more complicated dual structure

Hahn-Banach Theorem and Applications

• Hahn-Banach extension theorem allows extending bounded linear functionals from a subspace to the whole space
  • Preserves the norm of the functional during extension
• Hahn-Banach separation theorem asserts existence of separating hyperplanes between disjoint convex sets in a normed space
  • Fundamental tool in convex analysis and optimization
• Proves existence of non-trivial continuous linear functionals on any normed space
  • Guarantees rich dual structure for studying the space
• Allows characterizing the dual of certain function spaces ($L^p, C[a,b]$) via Riesz representation theorems
• Used in the proof of the open mapping and closed graph theorems in functional analysis
• Plays a crucial role in the development of the weak and weak* topologies on normed spaces

Weak and Weak* Topologies

• Weak topology on a normed space $X$ generated by the family of seminorms $\{p_f(x) = |f(x)|: f \in X^*\}$
  • Coarsest topology making all elements of the dual space $X^*$ continuous
• Weak* topology on the dual space $X^*$ generated by the family of seminorms $\{p_x(f) = |f(x)|: x \in X\}$
  • Coarsest topology making all evaluation functionals $e_x(f) = f(x)$ continuous
• Weak and weak* topologies generally coarser (fewer open sets) than the norm topology
  • Convergence in norm implies weak and weak* convergence, but not conversely
• Banach-Alaoglu theorem states that the closed unit ball of $X^*$ is compact in the weak* topology
  • Crucial tool for proving existence results and studying the dual space
• Weak and weak* topologies play a central role in the study of reflexive spaces and their properties

Reflexive Spaces and Their Properties

• Reflexive space $X$ isomorphic to its double dual $X^{**}$ via the canonical map $J: X \to X^{**}$ defined by $J(x)(f) = f(x)$
  • $J$ always injective, reflexivity equivalent to $J$ being surjective
• Reflexive spaces enjoy many desirable properties, such as the Radon-Nikodym property and the Krein-Milman theorem
• $\ell^p$ spaces reflexive for $1 < p < \infty$, while $\ell^1, c_0, L^1, C[a,b]$ are non-reflexive
  • Non-reflexivity often due to the presence of "singular" or "pathological" elements in the dual space
• Reflexivity preserved under isomorphisms, direct sums, and certain subspaces and quotients
• Kakutani's theorem characterizes reflexive spaces as those whose closed unit ball is weakly compact
• James' theorem states that a Banach space $X$ is reflexive if and only if every continuous linear functional on $X$ attains its norm

Examples and Counterexamples

• $\ell^p$ spaces ($1 \leq p \leq \infty$) serve as fundamental examples in the study of normed spaces and their duals
  • $\ell^p$ reflexive for $1 < p < \infty$, while $\ell^1$ and $\ell^\infty$ are non-reflexive
• $L^p$ spaces ($1 \leq p \leq \infty$) of $p$-integrable functions important in analysis and probability
  • $L^p$ reflexive for $1 < p < \infty$, while $L^1$ and $L^\infty$ are non-reflexive
• $C[a,b]$, the space of continuous functions on $[a,b]$, is a non-reflexive Banach space
  • Its dual is the space of signed Radon measures on $[a,b]$
• James' space $J$ a non-reflexive Banach space isomorphic to its double dual, but not reflexive
  • Shows that isomorphism to the double dual does not imply reflexivity
• Tsirelson's space $T$ a reflexive Banach space not containing any isomorphic copy of $\ell^p$ or $c_0$
  • Demonstrates the existence of "pathological" infinite-dimensional Banach spaces

Problem-Solving Techniques

• Understand the problem statement and the given information, such as the normed space, its dual, or specific properties
• Identify the relevant definitions, theorems, and properties that might be applicable to the problem
  • Examples: Hahn-Banach theorem, reflexivity, weak and weak* topologies, compactness
• Break down the problem into smaller, manageable parts or steps
  • Prove intermediate results or lemmas that can lead to the desired conclusion
• Use the properties of the specific space or its elements to simplify the problem or gain insights
  • Exploit the structure of $\ell^p, L^p, C[a,b]$, or other common spaces
• Consider using proof techniques such as contradiction, contraposition, or induction when appropriate
  • Indirect proofs can be useful when dealing with non-constructive existence results
• Visualize the problem geometrically, if possible, to gain intuition or guide the solution
  • Geometric interpretations of the Hahn-Banach theorem or the weak and weak* topologies can be helpful
• Check your solution for consistency, completeness, and correctness
  • Verify that all assumptions are used and that the conclusion follows logically from the arguments presented