10.4 Characterizations of reflexivity

Reflexivity in Banach spaces is a powerful concept with far-reaching implications. It's characterized by the attainment of norms on dual spaces and the coincidence of weak and weak* compactness. These properties provide valuable tools for analyzing linear functionals and bounded sets.

The Eberlein-Šmulian theorem further connects reflexivity to weak convergence of bounded sequences. This relationship highlights the importance of reflexivity in understanding the behavior of sequences and subsets in Banach spaces, making it a crucial concept in functional analysis.

Characterizations of Reflexivity

James' theorem for reflexivity

• Characterizes reflexivity in terms of the attainment of the norm on the dual space
  • Banach space $X$ is reflexive if and only if every continuous linear functional on $X$ attains its norm
• Proof assuming $X$ is reflexive:
  • Let $f \in X^*$ be a continuous linear functional on $X$
  • Hahn-Banach theorem guarantees existence of $x^{**} \in X^{**}$ such that $x^{**}(f) = \|f\|$ and $\|x^{**}\| = 1$
  • Reflexivity of $X$ implies existence of $x \in X$ such that $x^{**}(g) = g(x)$ for all $g \in X^*$
  • In particular, $f(x) = x^{**}(f) = \|f\|$, so $f$ attains its norm at $x$
• Proof assuming every continuous linear functional on $X$ attains its norm:
  • Let $x^{**} \in X^{**}$ be a bounded linear functional on $X^*$
  • Define $f(x) = x^{**}(J(x))$, where $J: X \to X^{**}$ is the canonical embedding
  • $f \in X^*$ and by assumption, there exists $x_0 \in X$ such that $f(x_0) = \|f\|$
  • For any $g \in X^*$, $|x^{**}(g) - g(x_0)| = |x^{**}(g) - f(x_0)| \leq \|x^{**}\| \|g - J(x_0)\|$
  • Taking the supremum over all $g$ with $\|g\| \leq 1$ yields $\|x^{**} - J(x_0)\| \leq \|x^{**}\| \|J(x_0) - J(x_0)\| = 0$
  • Thus, $x^{**} = J(x_0)$, so $X$ is reflexive

Weak compactness and reflexivity

• Weakly compact subset $K$ of Banach space $X$: every sequence in $K$ has a weakly convergent subsequence
  • Weak convergence: sequence $(x_n)$ in $X$ converges weakly to $x \in X$ if $f(x_n) \to f(x)$ for all $f \in X^*$
• Reflexive spaces characterized by coincidence of weak and weak* compactness
  • In reflexive space, bounded set is weakly compact if and only if it is weakly* compact
• Weakly compact sets in Banach space are bounded and closed in norm topology
• In reflexive space, closed unit ball is weakly compact

Eberlein-Šmulian theorem in reflexivity

• Eberlein-Šmulian theorem: for Banach space $X$, the following are equivalent:
  1. $X$ is reflexive
  2. Every bounded sequence in $X$ has a weakly convergent subsequence
  3. Every bounded subset of $X$ is weakly relatively compact
• Proof (1 ⇒ 2): Assume $X$ is reflexive, let $(x_n)$ be bounded sequence in $X$
  • Reflexivity of $X$ implies closed unit ball $B_X$ is weakly compact
  • Sequence $(x_n / \|x_n\|)$ contained in $B_X$, so it has a weakly convergent subsequence
  • Corresponding subsequence of $(x_n)$ is also weakly convergent
• Proof (2 ⇒ 3): Assume every bounded sequence in $X$ has a weakly convergent subsequence
  • Let $A \subseteq X$ be bounded, $(x_n)$ be sequence in $A$
  • By assumption, $(x_n)$ has a weakly convergent subsequence, so $A$ is weakly relatively compact
• Proof (3 ⇒ 1): Assume every bounded subset of $X$ is weakly relatively compact
  • In particular, closed unit ball $B_X$ is weakly relatively compact
  • Since $B_X$ is also weakly closed, it is weakly compact
  • Thus, $X$ is reflexive

Reflexivity vs Radon-Nikodým property

• Banach space $X$ has Radon-Nikodým property (RNP) if for every finite measure space $(\Omega, \Sigma, \mu)$ and every bounded linear operator $T: L^1(\mu) \to X$, there exists a Bochner integrable function $g: \Omega \to X$ such that $T(f) = \int_\Omega f g d\mu$ for all $f \in L^1(\mu)$
• Every reflexive Banach space has RNP
  • Proof: Let $X$ be reflexive, $T: L^1(\mu) \to X$ be bounded linear operator
  • Define bounded linear functional $\varphi$ on $L^1(\mu) \otimes X^*$ by $\varphi(f \otimes x^*) = \langle T(f), x^* \rangle$
  • Hahn-Banach theorem extends $\varphi$ to bounded linear functional on $L^1(\mu, X^*)$
  • Reflexivity of $X$ implies $L^1(\mu, X^*) \cong L^1(\mu, X)^*$, so there exists $g \in L^1(\mu, X)$ such that $\varphi(F) = \int_\Omega F(g) d\mu$ for all $F \in L^1(\mu, X^*)$
  • In particular, $\langle T(f), x^* \rangle = \int_\Omega f \langle g, x^* \rangle d\mu$ for all $f \in L^1(\mu)$ and $x^* \in X^*$
  • Thus, $T(f) = \int_\Omega f g d\mu$ for all $f \in L^1(\mu)$, so $X$ has RNP
• There exist non-reflexive Banach spaces with RNP ($\ell^1$)
• There exist reflexive Banach spaces without RNP ($L^1[0,1]$)