9.3 Weak and weak* convergence

Weak and weak* convergence are crucial concepts in functional analysis, offering alternative notions of convergence in normed spaces. They allow us to study sequences that don't converge in the usual norm topology but still exhibit useful convergence properties.

These convergence types are weaker than norm convergence but stronger than pointwise convergence. They're especially useful in infinite-dimensional spaces, where norm convergence can be too restrictive for many important sequences or series.

Weak and Weak* Convergence

Weak vs weak* convergence

• Weak convergence applies to sequences in a normed space $X$ converging to an element in $X$
  • Sequence $(x_n)$ converges weakly to $x$ if for every bounded linear functional $f$ in the dual space $X^*$, the sequence of scalars $f(x_n)$ converges to $f(x)$
  • Notation: $x_n \rightharpoonup x$
• Weak* convergence applies to sequences in the dual space $X^*$ converging to an element in $X^*$
  • Sequence $(f_n)$ converges weak* to $f$ if for every $x$ in the original space $X$, the sequence of scalars $f_n(x)$ converges to $f(x)$
  • Notation: $f_n \stackrel{*}{\rightharpoonup} f$
  • Only applicable in the dual space, not in the original normed space

Boundedness from weak convergence

• Weakly convergent sequences in a normed space are always bounded
  • Consequence of the Uniform Boundedness Principle
• If $(x_n)$ converges weakly to $x$, then there exists $M > 0$ such that $\|x_n\| \leq M$ for all $n$
  • If $(x_n)$ were unbounded, there would exist a functional $f$ such that $|f(x_n)|$ is unbounded, contradicting weak convergence
• Boundedness is a necessary condition for weak convergence, but not sufficient
  • Bounded sequences need not converge weakly ($\ell^\infty$ with coordinate functionals)

Weak* and pointwise convergence

• Weak* convergence of a sequence of functionals implies pointwise convergence
  • If $(f_n)$ converges weak* to $f$ in $X^*$, then for every fixed $x$ in $X$, the sequence of scalars $f_n(x)$ converges to $f(x)$
• Pointwise convergence does not imply weak* convergence
  • Pointwise convergence only considers convergence at each fixed point, while weak* convergence requires uniform convergence on bounded sets
• Weak* convergence is stronger than pointwise convergence
  • Weak* convergence guarantees pointwise convergence, but not vice versa ($\ell^1$ with coordinate functionals)

Examples of weak convergence types

• Weak convergence without norm convergence:
  • Sequence $(e_n)$ in $\ell^2$, where $e_n = (0, \ldots, 0, 1, 0, \ldots)$ with $1$ in the $n$-th position
    • Converges weakly to $0$ since for any $f \in (\ell^2)^*$, $\lim_{n \to \infty} f(e_n) = 0$
    • Does not converge in norm to $0$ since $\|e_n\| = 1$ for all $n$
• Weak* convergence without norm convergence:
  • Sequence $(f_n)$ in $\ell^1$, where $f_n(x) = \sum_{k=1}^n x_k$ for $x = (x_k) \in \ell^\infty$
    • Converges weak* to $f(x) = \sum_{k=1}^\infty x_k$ since for any $x \in \ell^\infty$, $\lim_{n \to \infty} f_n(x) = f(x)$
    • Does not converge in norm since $\|f_n\| = n$ for all $n$