Weak Convergence and the Banach-Alaoglu Theorem

Overview

Weak convergence is a notion of convergence for sequences in normed spaces that is weaker than norm (strong) convergence: a sequence (x_n) converges weakly to x if f(x_n) -> f(x) for every bounded linear functional f. The Banach-Alaoglu theorem states that the closed unit ball of the dual X* is compact in the weak* topology. These notions are essential in the calculus of variations, optimisation, and PDE theory, where minimising sequences often converge only weakly.

Intuition

Strong convergence (in norm) means x_n gets close to x in every direction. Weak convergence means every 'measurement' (bounded linear functional) of x_n converges to the measurement of x — but x_n itself need not get close to x in norm. For example, in L^2[0,1], the functions sin(nx) converge weakly to 0 (their averages against any fixed L^2 function tend to 0 by Riemann-Lebesgue), but ||sin(nx)||_2 = 1/sqrt(2) for all n. Weak convergence is useful because it is easier to achieve, and in reflexive spaces bounded sequences always have weakly convergent subsequences.

Formal Definition

Definition

Let X be a normed space. A sequence (x_n) converges weakly to x in X, written x_n -> x weakly or x_n -w-> x, if f(x_n) -> f(x) for every f in X*. A sequence (f_n) in X* converges weak* to f in X*, written f_n -w*-> f, if f_n(x) -> f(x) for every x in X. The weak topology on X is the coarsest topology making all f in X* continuous. The weak* topology on X* is the coarsest making all evaluation maps x |-> f(x) continuous.

x_n \xrightarrow{w} x \iff f(x_n) \to f(x) \;\forall f \in X^*

Weak convergence

f_n \xrightarrow{w^*} f \iff f_n(x) \to f(x) \;\forall x \in X

Weak-star convergence

\overline{B}_{X^*} = \{f \in X^* : \|f\| \le 1\} \text{ is weak}^*\text{-compact}

Banach-Alaoglu theorem

Notation

Notation	Meaning
$x_n \xrightarrow{w} x$	x_n converges weakly to x
$f_n \xrightarrow{w^*} f$	f_n converges weak* to f
$\sigma(X, X^*)$	Weak topology on X
$\sigma(X^*, X)$	Weak* topology on X*

Theorems

Theorem 14.1: Banach-Alaoglu Theorem

Let X be a normed space. The closed unit ball B_{X^*} = {f in X* : ||f|| <= 1} is compact in the weak* topology on X*.

Theorem 14.2: Weak Sequential Compactness in Reflexive Spaces

A Banach space X is reflexive (J: X -> X** surjective) if and only if every bounded sequence in X has a weakly convergent subsequence.

Theorem 14.3: Norm Semicontinuity under Weak Convergence

If x_n -> x weakly in a normed space X, then ||x|| <= liminf ||x_n||. In particular, weak limits can have smaller norm than the sequence.

Theorem 14.4: Mazur's Theorem

If x_n -w-> x in a Banach space X, then x is in the norm-closure of the convex hull of {x_n : n >= 1}. In particular, weak limits are strong limits of convex combinations.

Worked Examples

1
We need to show: for every g in L^2[0,1], integral_0^1 sin(nx) g(x) dx -> 0 as n -> infinity.
2
This is exactly the Riemann-Lebesgue lemma applied to g in L^2 (hence L^1): the Fourier coefficients of g tend to 0.
$\int_0^1 \sin(nx)g(x)\,dx = \hat{g}(n) \to 0 \text{ by Riemann-Lebesgue}$
3
Note: ||sin(nx)||_2 = 1/sqrt(2) for all n, so the sequence does NOT converge strongly. This illustrates the strict difference between weak and strong convergence.

✓ Answer

sin(nx) -w-> 0 in L^2[0,1] by Riemann-Lebesgue, but ||sin(nx)||_2 = 1/sqrt(2) so it does not converge strongly.

Practice Problems

Mediumproof writing

Prove that if x_n -w-> x in a normed space, then (x_n) is bounded: sup_n ||x_n|| < infinity.

MediumMultiple choice

Which statement about weak and strong convergence is correct?

Strong convergence implies weak convergenceWeak convergence implies strong convergenceWeak and strong convergence are equivalent in all Banach spacesStrong convergence is the same as weak* convergence

Mediumfree response

State and explain Mazur's theorem. How does it relate weak and strong convergence in convex sets?

Common Mistakes

Common Mistake

Assuming weak convergence implies strong convergence.

In infinite-dimensional spaces, weak convergence is strictly weaker than strong convergence. A sequence can converge weakly but have its norms bounded away from the norm of the limit.

Common Mistake

Confusing weak and weak* convergence.

Weak convergence in X uses functionals from X*. Weak* convergence in X* uses elements of X. When X is reflexive, weak and weak* topologies on X* coincide. In general they differ.

Quiz

Historical Background

Weak convergence arose naturally in the study of Fourier series and calculus of variations in the early twentieth century. The systematic theory was developed by Banach, Hahn, and others in the 1920s-30s. The Banach-Alaoglu theorem was proved by Leonidas Alaoglu in 1940, generalising an earlier result of Banach. Weak compactness results became crucial in existence proofs for PDEs and optimisation problems (direct method of the calculus of variations).

1920s
Banach and Hahn develop the theory of weak convergence
Stefan Banach, Hans Hahn
1932
Banach's monograph systematises weak convergence
Stefan Banach
1940
Alaoglu proves the weak* compactness of the unit ball
Leonidas Alaoglu
1950s
Weak compactness applied to existence theory for PDEs by various authors

Summary

Weak convergence: x_n -w-> x iff f(x_n) -> f(x) for all f in X*. Strictly weaker than norm convergence in infinite dimensions.
Weak* convergence: f_n -w*-> f in X* iff f_n(x) -> f(x) for all x in X.
Banach-Alaoglu: the unit ball of X* is weak*-compact.
Reflexive Banach spaces (including Hilbert spaces and L^p for 1 < p < inf) have the property that bounded sequences have weakly convergent subsequences.
Mazur's theorem: closed convex sets are weakly closed; weak limits of sequences in a closed convex set lie in the set.

References

BookRudin, W. — Functional Analysis (2nd ed.), Chapter 3
BookBrezis, H. — Functional Analysis, Sobolev Spaces and PDEs, Chapter 3
WebsiteWikipedia: Weak topology