Course 4 Book 4A: Function Spaces and Sobolev SpacesChapter 6

Weak Convergence and the Direct Method

Recovering compactness weakly, and minimising energies.

22 min read · Updated Oct 2, 2026

Read with Brezis, chapter 3, "Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity" (the sections on weak and weak-* convergence of sequences, Banach–Alaoglu, reflexivity and Mazur's lemma; the topological generalities can be skimmed), or Kreyszig §4.8–4.9 (strong and weak convergence).

In this chapter · 7 sections

6.1Fine twins in shape-memory alloys
6.2Weak convergence
6.3Weak compactness
6.4Lower semicontinuity and convexity
6.5The direct method
6.6History
6.7Exercises

Riesz's lemma (4A.1 Banach Spaces and Bounded Operators) closed a door: in an infinite-dimensional space, bounded sequences need not have convergent subsequences. This chapter opens another. If convergence is tested only against continuous linear functionals, that is, by "measurements" rather than by distance, then bounded sequences in the good spaces do have convergent subsequences. This is weak convergence. It is weaker than norm convergence (the harmonics of a string converge weakly to $0$ while staying at norm $1$ ), but it is often enough, because the quantities one wants to minimise behave well under it.

The payoff is the direct method of the calculus of variations, the most widely used way to prove that something exists by minimising: take a minimising sequence, extract a weakly convergent subsequence, and show the limit is a minimiser. It needs two things, coercivity (to keep the sequence bounded) and weak lower semicontinuity (so the limit doesn't do worse than the sequence). Convexity supplies the second. When convexity fails, minimisers can fail to exist, and minimising sequences develop ever-finer oscillations, which is exactly what certain metal alloys do.

By the end of this chapter you will be able to:

define weak and weak-* convergence, and show that orthonormal sequences, fast oscillations and spreading bumps converge weakly to $0$ ;
prove that weakly convergent sequences are bounded and that the norm can only drop in a weak limit;
state Banach–Alaoglu, prove its sequential form for separable spaces, and use reflexivity to extract weakly convergent subsequences;
explain why convex functionals are weakly lower semicontinuous;
run the direct method, and recognise from an example why it fails without convexity.

Fine twins in shape-memory alloys

In the world Model Microstructure as a minimising sequence

Shape-memory alloys such as nickel–titanium can be bent at low temperature and spring back to their original shape when warmed. The effect comes from a phase transformation in the crystal lattice. At low temperature the material is martensite, which can exist in several variants (the same distorted lattice, in different orientations). A piece of martensite that must fit a given overall shape, for instance at an interface with the high-temperature phase, often does so by forming a laminate: thin alternating layers of two variants, so fine that only their average shape is seen at the scale of the specimen. In real samples the layers can be micrometres or less in thickness, and they are often layered again at a finer scale inside each layer.

John Ball and Richard James explained this in 1987 ("Fine phase mixtures as minimizers of energy", Archive for Rational Mechanics and Analysis) as a phenomenon of the calculus of variations. The elastic energy prefers the deformation gradient to lie in one of a few "wells" (the variants), but the required average shape lies between them. No single deformation achieves the minimum energy; instead a minimising sequence of deformations alternates between the wells on finer and finer scales, its energy tending to the infimum. The sequence does not converge in norm. It converges weakly, to the averaged deformation, and the averaged deformation does not have the minimal energy. The microstructure seen in the microscope is a snapshot of a minimising sequence, stopped at a finite fineness by the small energy cost of the interfaces between layers. The mathematics of this chapter explains when such oscillation is forced (when the energy is not convex) and what the limit remembers.

Figure 6.1. A laminate of two phases at three scales of refinement (schematic). In

L^2

the indicator of the dark phase converges weakly, but not strongly, to the constant

\tfrac12

: averages over any fixed region converge, but the function never settles down at any point.

Weak convergence

Definition 6.1 Weak and weak-* convergence

A sequence $(x_n)$ in a normed space $X$ converges weakly to $x$ , written $x_n \rightharpoonup x$ , if $\phi(x_n) \to \phi(x)$ for every $\phi \in X^*$ . A sequence $(\phi_n)$ in a dual space $X^*$ converges weak-* to $\phi$ if $\phi_n(x) \to \phi(x)$ for every $x \in X$ .

Norm convergence implies weak convergence ( $|\phi(x_n) - \phi(x)| \leq \|\phi\|\,\|x_n - x\|$ ). In finite dimensions the two coincide (test against the coordinate functionals). In infinite dimensions they don't.

Example 6.2 Weakly but not strongly

Orthonormal sequences. In a Hilbert space, an orthonormal sequence $(e_n)$ converges weakly to $0$ : by Riesz representation (4A.4 Hilbert Spaces and Lax–Milgram) a functional is $\langle\cdot, y\rangle$ , and $\sum|\langle e_n, y\rangle|^2 \leq \|y\|^2$ (Bessel) forces $\langle e_n, y\rangle \to 0$ . But $\|e_n\| = 1$ . The harmonics of 4A.1 Banach Spaces and Bounded Operators's string converge weakly to $0$ .
Oscillation. In $L^2(0, 2\pi)$ , $\sin(nx) \rightharpoonup 0$ : for $g \in L^2$ , $\int g(x)\sin(nx)\,dx \to 0$ (Riemann–Lebesgue). Fast vibration averages out against any fixed test function (Figure 6.2).
Escape. In $L^p(\mathbb{R})$ with $1 < p < \infty$ , the sliding bump $1_{[n, n+1]}$ , the spreading plateau $n^{-1/p}1_{[0, n]}$ and the concentrating spike $n^{1/p}1_{[0, 1/n]}$ , all of norm $1$ , converge weakly to $0$ (Exercise 6.10). The three escapes of 3A.3 The Lebesgue Integral are invisible to weak convergence.
Laminates. If $\chi_n(x) = \chi(nx)$ for a $1$ -periodic $\chi$ (on a bounded set), then $\chi_n \rightharpoonup \int_0^1\chi$ , the average (Exercise 6.11).

Figure 6.2.

\sin(nx)

for

n = 3, 10, 30

against a fixed test function

g

(dashed, a step function). The integral

\int g\sin(nx)

, plotted against

n

(below, computed), tends to

0

, like

1/n

: the oscillations cancel against any fixed

g

, although

\|\sin(nx)\|_2

stays constant.

Weak limits keep some properties of the sequence and lose others.

Proposition 6.3 Properties of weak limits

Let $x_n \rightharpoonup x$ in a normed space $X$ . Then:

the weak limit is unique;
$(x_n)$ is bounded;
$\|x\| \leq \liminf_n\|x_n\|$ ;
in a Hilbert space, if also $\|x_n\| \to \|x\|$ , then $x_n \to x$ in norm.

Proof. (1) If $x_n \rightharpoonup x$ and $x_n \rightharpoonup y$ , then $\phi(x - y) = 0$ for every $\phi$ , so $x = y$ (4A.3 Hahn–Banach and Duality, functionals separate points). (2) View each $x_n$ as a functional on the Banach space $X^*$ ; the family is pointwise bounded, so by uniform boundedness (4A.2 Baire Category and Its Consequences) it is bounded in norm, and the norm of $x_n$ as a functional is $\|x_n\|$ (4A.3 Hahn–Banach and Duality). (3) Take a norming functional $\phi$ for $x$ (4A.3 Hahn–Banach and Duality): $\|x\| = \phi(x) = \lim\phi(x_n) \leq \liminf\|x_n\|$ . (4) $\|x_n - x\|^2 = \|x_n\|^2 - 2\operatorname{Re}\langle x_n, x\rangle + \|x\|^2 \to \|x\|^2 - 2\|x\|^2 + \|x\|^2 = 0$ .

Property (3) is weak lower semicontinuity of the norm: in a weak limit, the norm can drop (as in all the examples, where it drops from $1$ to $0$ ) but never jump up. It is the same phenomenon as Fatou's lemma (3A.3 The Lebesgue Integral), and property (4) says that in a Hilbert space a drop in norm is the only way weak convergence can fail to be strong.

Weak compactness

Theorem 6.4 Banach–Alaoglu

The closed unit ball of the dual $X^*$ of a normed space is compact in the weak-* topology. If $X$ is separable, every bounded sequence in $X^*$ has a weak-* convergent subsequence.

The general statement needs topology beyond sequences (Tychonoff's theorem); the sequential version for separable $X$ , which is the one used in analysis, needs only the diagonal argument.

Proof. Let $\{x_1, x_2, \ldots\}$ be dense in $X$ and $(\phi_n)$ a sequence with $\|\phi_n\| \leq 1$ . The numbers $\phi_n(x_1)$ are bounded, so a subsequence converges; from it extract one along which $\phi_n(x_2)$ converges; and so on. The diagonal subsequence $\psi_k$ converges at every $x_j$ (2B.5 Uniform Convergence and Arzelà–Ascoli). Since $\|\psi_k\| \leq 1$ , the family is equicontinuous, and an $\varepsilon/3$ argument spreads convergence from the dense set to all $x$ : for $x \in X$ choose $x_j$ with $\|x - x_j\| < \varepsilon$ , then $|\psi_k(x) - \psi_l(x)| \leq 2\varepsilon + |\psi_k(x_j) - \psi_l(x_j)|$ . The limit $\phi(x) = \lim\psi_k(x)$ is linear with $\|\phi\| \leq 1$ .

The proof is Arzelà–Ascoli again: pointwise convergence on a countable dense set, then equicontinuity. For a reflexive space, $X = (X^*)^*$ , and weak-* convergence in $X^{**}$ is weak convergence in $X$ :

Corollary 6.5 Bounded sequences in reflexive spaces

In a reflexive Banach space, in particular in any Hilbert space and in $L^p$ for $1 < p < \infty$ , every bounded sequence has a weakly convergent subsequence.

(For a separable reflexive space this follows from Banach–Alaoglu applied to $X^*$ , which is then separable; the general case is the Eberlein–Šmulian theorem, Brezis Theorem 3.18. For a Hilbert space it can also be proved directly, by the diagonal argument on an orthonormal basis.)

This is the replacement for Bolzano–Weierstrass that analysis in infinite dimensions runs on. It fails in $L^1$ : the spikes $n1_{[0, 1/n]}$ are bounded in $L^1$ but have no weakly convergent subsequence in $L^1$ . Their mass concentrates, and the natural limit is the Dirac mass $\delta_0$ , which is a measure, not an $L^1$ function. Viewed in the larger space of measures, the dual of $C([0, 1])$ (4A.3 Hahn–Banach and Duality), Banach–Alaoglu gives weak-* convergence to $\delta_0$ . That is why problems with linear growth are hard, and why their solutions are sometimes measures.

Lower semicontinuity and convexity

A functional $E : X \to (-\infty, +\infty]$ is weakly lower semicontinuous if $x_n \rightharpoonup x$ implies $E(x) \leq \liminf E(x_n)$ . The norm is one; which others?

Theorem 6.6 Convex functionals are weakly lower semicontinuous

Let $X$ be a Banach space and $E : X \to (-\infty, +\infty]$ convex and lower semicontinuous for norm convergence. Then $E$ is weakly lower semicontinuous.

Proof. For each $c$ , the sublevel set $\{E \leq c\}$ is convex and norm-closed. A norm-closed convex set is weakly closed: if $x \notin C$ , the geometric Hahn–Banach theorem (4A.3 Hahn–Banach and Duality) gives $\phi$ and $\alpha$ with $\phi(x) > \alpha \geq \phi(y)$ for $y \in C$ , and then a sequence in $C$ can't converge weakly to $x$ . Now if $x_n \rightharpoonup x$ and $\liminf E(x_n) < c$ , then infinitely many $x_n$ lie in $\{E \leq c\}$ , so $x$ does too: $E(x) \leq c$ .

An equivalent formulation is Mazur's lemma: a weak limit is a norm limit of convex combinations of the sequence's terms. Convexity is exactly what allows averaging, and weak limits are averages.

Typical convex functionals in PDE are the energies $E(u) = \int\big(\tfrac12|\nabla u|^2 - fu\big)$ , or more generally $\int F(x, u, \nabla u)$ with $F$ convex in $\nabla u$ . Nonlinear functions of $u$ that are not convex behave badly under weak limits: if $u_n \rightharpoonup u$ , then $\int u_n^2$ need not converge to $\int u^2$ (it can only drop), and $\int\frac1{a_n}$ need not converge to $\int\frac1a$ , which is the homogenisation phenomenon below.

In the world Model The conductivity of a layered material

A composite made of thin parallel layers of two materials, with conductivities $a_1$ and $a_2$ in equal proportions, conducts differently along and across the layers. Along the layers, the materials carry current side by side and the effective conductivity is the arithmetic mean $\frac{a_1 + a_2}2$ . Across the layers, the current must pass through each in series, the resistances $\frac1{a_i}$ add, and the effective conductivity is the harmonic mean $\big(\frac{1/a_1 + 1/a_2}{2}\big)^{-1}$ . For $a_1 = 1$ and $a_2 = 9$ these are $5$ and $1.8$ , very different.

In the language of this chapter: the rapidly oscillating conductivity $a_n(x) = a(nx)$ converges weakly to its average, and so does $\frac{1}{a_n}$ , to the average of $\frac1a$ , but the weak limit of $\frac1{a_n}$ is not $\frac{1}{\text{weak limit of }a_n}$ . Which average appears in the effective equation depends on how the oscillation is oriented relative to the current. This is the starting point of homogenisation theory, which computes effective properties of fine-scale composites, and the harmonic and arithmetic means are the extreme possible values (the Wiener bounds).

The direct method

Theorem 6.7 The direct method

Let $X$ be a reflexive Banach space and $E : X \to (-\infty, +\infty]$ , not identically $+\infty$ , be

coercive: $E(x) \to +\infty$ as $\|x\| \to \infty$ , and
weakly lower semicontinuous.

Then $E$ attains its infimum. If $E$ is strictly convex, the minimiser is unique.

Proof. Let $m = \inf E < +\infty$ and $(x_n)$ a minimising sequence, $E(x_n) \to m$ . By coercivity $(x_n)$ is bounded (otherwise a subsequence would have $E \to \infty$ ). By Corollary 6.5 a subsequence converges weakly, to some $x$ . By lower semicontinuity $E(x) \leq \liminf E(x_{n_k}) = m$ , so $E(x) = m$ (in particular $m > -\infty$ ). If $x \neq y$ were both minimisers and $E$ strictly convex, $E(\frac{x + y}2) < m$ .

Figure 6.3. The direct method as a chain of steps. It has the same shape as the contradiction–compactness template of 2B.3 Compactness: a sequence, a bound, a compactness theorem, and passing to the limit. Here the compactness is weak, so the last step needs lower semicontinuity rather than continuity.

The Dirichlet principle of 4A.4 Hilbert Spaces and Lax–Milgram is the simplest instance: $E(u) = \int(\tfrac12|\nabla u|^2 - fu)$ on $H^1_0$ is strictly convex, continuous and (by the Poincaré inequality, 4A.9 Sobolev Spaces) coercive, so it has a unique minimiser, the weak solution of $-\Delta u = f$ . The method's real strength is for nonlinear problems where no linear theory applies: minimal surfaces and harmonic maps (6A.9 Calculus of Variations and Gradient Flows), and the minimisers behind Perelman's $\mu$ -functional (12A.3 The 𝓦-Entropy).

Example 6.8 When convexity fails: no minimiser

Let $E(u) = \int_0^1\big((u'^2 - 1)^2 + u^2\big)\,dx$ over functions with $u(0) = u(1) = 0$ and $u' \in L^4$ (Bolza's example). The integrand prefers slopes $u' = \pm1$ , which a zigzag with $n$ teeth and height $\frac1{2n}$ achieves, while keeping $u$ small: its energy is $\int u^2 \leq \frac1{4n^2} \to 0$ . So $\inf E = 0$ . But $E(u) = 0$ would require $u = 0$ and $|u'| = 1$ at once, which is impossible: there is no minimiser. The zigzags converge weakly (in fact uniformly) to $0$ , whose energy is $E(0) = 1$ , more than the limit $0$ of the energies. Lower semicontinuity fails, because $(p^2 - 1)^2$ is not convex in $p$ . The minimising sequence develops ever-finer oscillations between the two preferred slopes: the one-dimensional cartoon of the martensite laminate.

Where this goes The direct method in the rest of the guidebook

Elliptic equations and harmonic maps (6A.9 Calculus of Variations and Gradient Flows): convex energies on Sobolev spaces, minimised directly.
Eigenvalues (4A.7 Compact Operators and Spectra): the first eigenvalue of $-\Delta$ on a bounded domain is the minimum of the Rayleigh quotient, attained because Rellich's theorem (4A.10 Sobolev Embeddings and Critical Exponents) upgrades weak to strong convergence in $L^2$ .
Perelman's $\lambda$ and $\mu$ (12A.2 Ricci Flow as a Gradient Flow, 12A.3 The 𝓦-Entropy): on a closed manifold, $\lambda(g) = \inf\int(4|\nabla w|^2 + Rw^2)$ over $\int w^2 = 1$ is a minimum by exactly the eigenvalue argument. For $\mu$ , the functional includes $-\int w^2\log w^2$ , which is not convex, and the proof that a minimiser exists uses the log-Sobolev inequality to control it (6A.10 Entropy, Information and Diffusion); on non-compact manifolds a minimising sequence can escape to infinity, and minimisers can fail to exist, so the direct method must be applied with care.
Min–max (10A.8 Min–Max and Width, 12C.2 Finite Extinction): when the object sought is a saddle rather than a minimum, minimising sequences are replaced by "sweepouts", but the same issues of compactness and lower semicontinuity decide whether a limit exists.

History

Hilbert used weak convergence ("schwache Konvergenz") in his work on integral equations around 1906. Banach's 1932 book proved the sequential compactness theorem for duals of separable spaces, and Leonidas Alaoglu the general theorem in 1940. Stanisław Mazur's lemma dates from 1933. The direct method was Hilbert's response, in 1900, to Weierstrass's criticism of Riemann's use of the Dirichlet principle; Leonida Tonelli developed lower semicontinuity as its central tool in the 1910s and 1920s. Oskar Bolza's example of non-existence comes from the early calculus of variations. Ball and James's analysis of martensitic microstructure appeared in 1987.

Recall Where we stand

Weak convergence tests sequences against functionals. Orthonormal sequences, oscillations and the three escapes converge weakly to $0$ but not strongly. Weak limits are unique, weakly convergent sequences are bounded, and the norm can only drop in a weak limit (in a Hilbert space, a drop is the only obstruction to strong convergence). Banach–Alaoglu gives weak-* compactness of dual balls, and in reflexive spaces bounded sequences have weakly convergent subsequences. Convex, norm-lower-semicontinuous functionals are weakly lower semicontinuous, and the direct method then produces minimisers of coercive functionals; without convexity, minimising sequences can oscillate and minimisers can fail to exist. 4A.7 Compact Operators and Spectra studies compact operators, which turn weak convergence into strong, and proves the spectral theorem.

Exercises

Exercise 6.9 Oscillation

(a) Show that $\sin(nx) \rightharpoonup 0$ in $L^2(0, 2\pi)$ but not in norm. (b) Show that $\sin^2(nx) \rightharpoonup \tfrac12$ . So weak limits don't commute with squaring: the weak limit of the squares is not the square of the weak limit.

Solution

(a) For a step function $g$ , $\int g\sin(nx) = O(1/n)$ ; step functions are dense in $L^2$ and $\|\sin(nx)\|_2 = \sqrt\pi$ is bounded, so an $\varepsilon/2$ argument extends to all $g$ . Not in norm: $\|\sin(nx)\|_2 = \sqrt\pi$ . (b) $\sin^2(nx) = \tfrac12 - \tfrac12\cos(2nx)$ , and $\cos(2nx) \rightharpoonup 0$ likewise.

Exercise 6.10 Escapes are weakly invisible

For $1 < p < \infty$ and conjugate $q$ , show that $1_{[n, n+1]}$ , $n^{-1/p}1_{[0, n]}$ and $n^{1/p}1_{[0, 1/n]}$ converge weakly to $0$ in $L^p(\mathbb{R})$ . (Test against $g \in L^q$ ; use Hölder and the fact that $\int_E|g|^q \to 0$ when $E$ shrinks or moves to infinity, and for the plateau approximate $g$ by a compactly supported function.) Why does the spike fail to converge weakly in $L^1$ ?

Exercise 6.11 Periodic oscillations average out

Let $\chi$ be bounded, measurable and $1$ -periodic on $\mathbb{R}$ , with average $\bar\chi = \int_0^1\chi$ . Show that $\chi(nx) \rightharpoonup \bar\chi$ in $L^2(0, 1)$ . (Check against indicators of intervals first, then use density and boundedness.)

Exercise 6.12 Bounded below, no minimiser

On $\ell^2$ let $E(x) = (\|x\|^2 - 1)^2 + \sum_k\frac{x_k^2}{k}$ . Show $\inf E = 0$ but $E > 0$ everywhere. Which hypothesis of the direct method fails? (Evaluate $E$ on the basis vectors $e_n$ .)

Solution

$E(e_n) = \frac1n \to 0$ . $E(x) = 0$ would need $\|x\| = 1$ and all $x_k = 0$ . $E$ is coercive, but not weakly lower semicontinuous: $e_n \rightharpoonup 0$ and $E(0) = 1 > 0 = \lim E(e_n)$ . The term $(\|x\|^2 - 1)^2$ is not convex.

Exercise 6.13 The two means

For layers of conductivities $a_1, a_2$ in proportions $\theta$ and $1 - \theta$ , compute the effective conductivities along and across the layers, and show the harmonic mean never exceeds the arithmetic mean, with equality only if $a_1 = a_2$ (3A.7 Lᵖ Spaces and Jensen’s Inequality's Jensen).

Exercise 6.14 Bolza's example in detail

For the zigzag $u_n$ with slope $\pm1$ , $n$ teeth and zero boundary values, compute $\int_0^1u_n^2$ exactly and show $E(u_n) \to 0$ . Show that $u_n \to 0$ uniformly and $u_n' \rightharpoonup 0$ weakly in $L^4$ .

Solution

Each tooth on an interval of length $\frac1n$ is a triangle of height $\frac1{2n}$ , with $\int u^2 = 2\int_0^{1/(2n)}t^2\,dt = \frac{1}{12n^3}$ ; $n$ teeth give $\frac{1}{12n^2}$ . Since $(u_n'^2 - 1)^2 = 0$ , $E(u_n) = \frac{1}{12n^2}$ . $|u_n| \leq \frac1{2n}$ . And $u_n'$ is a $\frac1n$ -periodic $\pm1$ function with average $0$ , so it converges weakly to $0$ (Exercise 6.11).

Exercise 6.15 Rehearsal: a minimising sequence that escapes

On $\mathbb{R}^n$ , let $R(u) = \int|\nabla u|^2\big/\int u^2$ for non-zero $u \in H^1(\mathbb{R}^n)$ . Using $u_\lambda(x) = u(\lambda x)$ , show $R(u_\lambda) = \lambda^2R(u)$ , so $\inf R = 0$ , and that the infimum is not attained. A minimising sequence $u_\lambda$ , $\lambda \to 0$ , normalised in $L^2$ , spreads out: it is escape to width infinity (3A.3 The Lebesgue Integral), and it converges weakly to $0$ . On a closed manifold this can't happen (Rellich, 4A.10 Sobolev Embeddings and Critical Exponents), which is why Perelman's $\lambda(g)$ is attained there. On non-compact manifolds the same failure must be ruled out by other means, a recurring technical issue in 12A.3 The 𝓦-Entropy and in the study of κ-solutions (12B.1 κ-Solutions).

Solution

$\nabla(u_\lambda) = \lambda(\nabla u)(\lambda x)$ , so the numerator scales by $\lambda^2\lambda^{-n}$ and the denominator by $\lambda^{-n}$ . If $R(u) = 0$ then $\nabla u = 0$ , so $u$ is constant, and in $L^2(\mathbb{R}^n)$ that means $u = 0$ .

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6.1Fine twins in shape-memory alloys

6.2Weak convergence

6.3Weak compactness

6.4Lower semicontinuity and convexity

6.5The direct method

6.6History

6.7Exercises

Fine twins in shape-memory alloys

Weak convergence

Weak compactness

Lower semicontinuity and convexity

The direct method

History

Exercises