In finite-dimensional spaces like $\mathbb{R}^n$, the Bolzano-Weierstrass theorem guarantees that every bounded sequence has a convergent subsequence. This sequential compactness is the engine behind most existence arguments in analysis: to find a minimiser of a functional, one extracts a limit from a minimising sequence and checks it solves the problem. In infinite-dimensional Banach spaces — the natural setting for PDEs — this completely breaks down. The closed unit ball of an infinite-dimensional Banach space is never compact in the norm topology, and a bounded sequence can oscillate indefinitely without accumulating anywhere.

The remedy is to work with a coarser topology, called the weak topology, in which far more sequences converge. Rather than demanding that $x_n$ approach $x$ in norm, we only ask that every continuous linear functional evaluates to the right limit. This is precisely weak enough to restore compactness (in reflexive spaces) while remaining strong enough to pass to limits in the linear equations arising in PDE theory.

Motivation

[motivation]

The Compactness Crisis in Infinite Dimensions

The Bolzano-Weierstrass theorem states that every bounded sequence in $\mathbb{R}^n$ has a convergent subsequence. The proof relies on the Heine-Borel property: closed bounded subsets of $\mathbb{R}^n$ are compact. In infinite-dimensional Banach spaces, Heine-Borel fails catastrophically. The following result, due to Riesz, quantifies the failure: if $Y$ is a proper closed subspace of a Banach space $X$, then for every $\varepsilon > 0$ there exists $x \in X$ with $\|x\|_X = 1$ and $\mathrm{dist}(x, Y) > 1 - \varepsilon$. Iterating this with a nested sequence of subspaces produces a sequence in the closed unit ball $\overline{B}(0, 1) \subseteq X$ with $\|x_m - x_n\|_X > 1/2$ for all $m \neq n$. Such a sequence has no convergent subsequence. The unit ball is therefore not (sequentially) compact in the norm topology of any infinite-dimensional Banach space.

This is not a pathology — it is the generic situation in PDE theory. The natural spaces $L^p(U)$, $W^{k,p}(U)$, and $C^k(\overline{U})$ are all infinite-dimensional, and every bounded sequence of approximate solutions lives in a non-compact ball.

What Goes Wrong With Minimising Sequences

The loss of compactness directly obstructs the most natural existence strategy: the direct method. To solve a variational problem $\inf_{u \in A} I(u)$ for some functional $I$ and admissible set $A$, one constructs a minimising sequence $\{u_n\}$ with $I(u_n) \to \inf I$. The energy bound $I(u_n) \leq C$ typically implies $\|u_n\|_X \leq C'$ for some Banach norm, but without compactness of the ball, there is no reason for $\{u_n\}$ to have a norm-convergent subsequence. Even if the functional $I$ is well-behaved, the argument stalls because we cannot produce a candidate limit.

The Fix: Test With Functionals

The insight is to weaken what "convergence" means. Instead of requiring $\|x_n - x\| \to 0$ — which asks the sequence to be close in every direction simultaneously — we only ask that $f(x_n) \to f(x)$ for each continuous linear functional $f \in X^*$ separately. This is a strictly weaker condition: each functional tests the sequence along a single "direction" in the dual, and we demand convergence one direction at a time rather than uniformly. The resulting notion — weak convergence — admits far more convergent sequences. In reflexive Banach spaces (which include all $L^p$ and $W^{k,p}$ spaces for $1 < p < \infty$), every bounded sequence has a weakly convergent subsequence, restoring the compactness that the direct method requires.

The price is that weak limits are harder to work with: the norm can drop, nonlinear expressions need not pass to the limit, and strong convergence must be recovered separately when needed. Understanding this trade-off — what passes to the weak limit and what does not — is the core of the theory.

[/motivation]

Weak Convergence

[definition:Weak Topology]
Let $X$ be a real Banach space with continuous dual $X^*$, consisting of all bounded linear functionals
\begin{align*} f: X &\to \mathbb{R} \\ x &\mapsto f(x). \end{align*}
The weak topology on $X$ is the coarsest topology under which every evaluation map $x \mapsto f(x)$ is continuous, for all $f \in X^*$.
[/definition]

The weak topology is strictly coarser than the norm topology whenever $X$ is infinite-dimensional: every weakly open set is norm-open (since each $f \in X^*$ is norm-continuous), but not conversely. The open unit ball $B(0,1)$ is norm-open but not weakly open — every weakly open neighbourhood of the origin contains a translate of a closed subspace of finite codimension, and therefore contains points of arbitrarily large norm.

The weak topology is an initial topology — the coarsest making a specified family of maps continuous — so its convergence behaviour is completely determined by the family. The following theorem makes this precise and provides the sequential criterion that is used in practice. We write $x_n \rightharpoonup x$ to denote weak convergence.

[quotetheorem:255]

The forward direction is immediate from continuity: if $x_n \to x$ in the weak topology, then each $f \in X^*$ (being weakly continuous by construction) maps the convergent sequence to a convergent sequence. The reverse direction uses the fact that every weak neighbourhood of $x$ contains a finite intersection of sub-basic sets $\{y : |f_i(y) - f_i(x)| < \varepsilon\}$ for finitely many $f_1, \ldots, f_m \in X^*$, and pointwise convergence in each $f_i$ ensures $x_n$ eventually enters the intersection. The theorem reduces checking convergence in the (abstractly defined) weak topology to checking countably many real-number limits — one for each $f \in X^*$.

Strong (norm) convergence always implies weak convergence: if $\|x_n - x\|_X \to 0$ then $|f(x_n) - f(x)| \leq \|f\|_{X^*}\|x_n - x\|_X \to 0$ for every $f \in X^*$, by the definition of the operator norm. The converse is false in general — the oscillation example below exhibits a sequence that converges weakly to zero while maintaining unit norm.

Weak Convergence in $L^p$ Spaces

In the abstract setting, weak convergence tests against an arbitrary $f \in X^*$. In $L^p$ spaces for $1 < p < \infty$, the Riesz Representation Theorem identifies the dual $L^p(U)^*$ isometrically with $L^q(U)$ (where $1/p + 1/q = 1$), via the pairing $f_v(u) = \int_U u \, v \, d\mathcal{L}^n$ for $v \in L^q(U)$. Under this identification, weak convergence $u_n \rightharpoonup u$ in $L^p(U)$ is equivalent to the integral condition
\begin{align*} \lim_{n \to \infty} \int_U u_n(x)\, v(x) \, d\mathcal{L}^n(x) = \int_U u(x)\, v(x) \, d\mathcal{L}^n(x) \quad \text{for every } v \in L^q(U). \end{align*}
This concrete characterisation is what is used in practice: to check weak convergence in $L^p$, one tests the sequence against every function in the conjugate space $L^q$, and "convergence" means that every such integral converges.

The case $p = 1$ is subtler. The dual of $L^1(U)$ is $L^\infty(U)$, so weak convergence in $L^1$ requires $\int u_n v \, d\mathcal{L}^n \to \int u v \, d\mathcal{L}^n$ for all $v \in L^\infty$. This is a strictly stronger condition than convergence of integrals against continuous functions alone, and the failure of reflexivity for $L^1$ means that bounded sequences in $L^1$ need not have weakly convergent subsequences (see Problem 1).

Weak Sequential Compactness

The central compactness theorem for weak convergence restores, in reflexive spaces, the bounded-implies-subsequentially-convergent property that fails in the norm topology.

[quotetheorem:214]

The hypothesis of reflexivity is essential. A Banach space $X$ is reflexive if and only if the canonical isometric embedding $J: X \hookrightarrow X^{**}$ defined by $J(x)(f) := f(x)$ is surjective — that is, every element of the bidual is already an evaluation functional at some point of $X$. The proof of the theorem works by applying the Banach-Alaoglu theorem to the sequence $\{J(x_n)\}$ in $X^{**}$ (which lies in a weak*-compact ball) and then using surjectivity of $J$ to pull the limit back to $X$. The class of reflexive spaces covers the most important cases in PDE theory: $L^p(U)$ and $W^{k,p}(U)$ are reflexive for $1 < p < \infty$. The theorem fails for $L^1(U)$, $L^\infty(U)$, and $C(\overline{U})$, none of which are reflexive — bounded sequences in these spaces can concentrate, oscillate, or escape to the boundary without converging weakly.

Lower Semicontinuity of the Norm

[quotetheorem:215]

The inequality $\|x\| \leq \liminf \|x_n\|$ is typically strict: weak limits can have strictly smaller norm than the approximating sequence. Physically, this corresponds to energy being lost to oscillations or concentrations that the weak limit does not capture. The proof is a one-line application of the Hahn-Banach theorem: choose a norming functional $f \in X^*$ with $\|f\|_{X^*} = 1$ and $f(x) = \|x\|_X$, then $\|x\|_X = \lim f(x_n) \leq \liminf \|x_n\|_X$. Lower semicontinuity is what makes the direct method work: even though the norm (or energy) can drop in a weak limit, it cannot increase, so the infimum is preserved.

An important upgrade is available: if $x_n \rightharpoonup x$ and $\|x_n\|_X \to \|x\|_X$ (i.e. the norm does not drop), then $x_n \to x$ strongly. In Hilbert spaces this follows immediately from expanding $\|x_n - x\|^2 = \|x_n\|^2 - 2\operatorname{Re}\langle x_n, x \rangle + \|x\|^2$ and noting that weak convergence gives $\langle x_n, x \rangle \to \|x\|^2$. In uniformly convex Banach spaces (which include $L^p$ for $1 < p < \infty$), the same conclusion holds by the Milman-Pettis theorem and the Kadec-Klee property. This gives a practical strategy for upgrading weak to strong convergence: prove the energy converges to the right value.

Mazur's Lemma

[quotetheorem:216]

Mazur's lemma is a bridge between weak and strong convergence: it says that although the sequence $\{x_n\}$ itself may not converge strongly, one can always build convex combinations of its tails that do. The proof is a clean application of the Hahn-Banach separation theorem: if $x$ were not in the norm-closure of the convex hull of $\{x_k : k \geq n\}$, a separating functional would contradict weak convergence. No additional structure (reflexivity, uniform convexity) is needed — the result holds in every Banach space.

The lemma is particularly useful in the calculus of variations and lower semicontinuity arguments. Suppose a functional $I$ is known to be strongly lower semicontinuous and convex, and one has a minimising sequence $u_n \rightharpoonup u$. Mazur's lemma provides convex combinations $v_n \to u$ strongly. Then $I(v_n) \leq \max_k I(u_{n_k})$ by convexity, and the strong lower semicontinuity gives $I(u) \leq \liminf I(v_n) \leq \liminf I(u_n)$, recovering the weak lower semicontinuity of $I$ from its strong lower semicontinuity and convexity. This is the key step in proving that convex integral functionals are weakly lower semicontinuous — the foundation of the direct method in the Calculus of Variations.

Examples and Counterexamples

Oscillation: Weak but Not Strong

The canonical illustration of weak-but-not-strong convergence is high-frequency oscillation, where the sequence disperses its energy across increasingly fine scales.

[example: Oscillation in $L^2$]
Consider $H = L^2(0, 2\pi)$ with the standard inner product, and define
\begin{align*} u_n: (0, 2\pi) &\to \mathbb{R} \\ x &\mapsto \sin(nx). \end{align*}
We claim $u_n \rightharpoonup 0$ weakly but $u_n \not\to 0$ strongly.

Failure of strong convergence. A direct computation gives
\begin{align*} \|u_n\|_{L^2}^2 = \int_0^{2\pi} \sin^2(nx) \, d\mathcal{L}^1(x) = \int_0^{2\pi} \frac{1 - \cos(2nx)}{2} \, d\mathcal{L}^1(x) = \pi, \end{align*}
so $\|u_n\|_{L^2} = \sqrt{\pi} \neq 0$ for all $n$, and the sequence cannot converge strongly to zero.

Weak convergence. Let $v \in L^2(0, 2\pi)$. For any $\varepsilon > 0$, density of smooth functions provides $\phi \in C_c^\infty(0, 2\pi)$ with $\|v - \phi\|_{L^2} < \varepsilon$. We split:
\begin{align*} \left|\int_0^{2\pi} u_n v \, d\mathcal{L}^1\right| \leq \left|\int_0^{2\pi} u_n \phi \, d\mathcal{L}^1\right| + \|u_n\|_{L^2}\|v - \phi\|_{L^2} \leq \left|\int_0^{2\pi} u_n \phi \, d\mathcal{L}^1\right| + \sqrt{\pi}\,\varepsilon. \end{align*}
Integrating by parts (boundary terms vanish since $\phi$ has compact support),
\begin{align*} \int_0^{2\pi} \sin(nx)\, \phi(x) \, d\mathcal{L}^1(x) = \frac{1}{n}\int_0^{2\pi} \cos(nx)\, \phi'(x) \, d\mathcal{L}^1(x), \end{align*}
which is bounded by $\frac{1}{n}\|\phi'\|_{L^1} \to 0$. Since $\varepsilon$ was arbitrary, the full integral tends to zero, confirming $u_n \rightharpoonup 0$.

The norm drops strictly: $\|u_n\|_{L^2} = \sqrt{\pi}$ for all $n$ while $\|0\|_{L^2} = 0$. This is consistent with the Lower Semicontinuity of the Norm, which only guarantees $\|x\| \leq \liminf \|x_n\|$. The energy $\pi$ per period is not lost — it is redistributed to ever-higher frequencies, where no single $L^2$ test function can detect it.
[/example]

The Geometry of Weak Convergence

The oscillation example illustrates a general geometric principle. In a Hilbert space with orthonormal basis $\{e_n\}$, the sequence $e_n$ converges weakly to zero: for any $x \in H$, the inner product $\langle e_n, x \rangle$ is the $n$-th Fourier coefficient of $x$, which tends to zero by Bessel's inequality and convergence of $\sum |\langle e_n, x \rangle|^2$. Yet $\|e_n - e_m\| = \sqrt{2}$ for $n \neq m$, so the sequence has no strongly convergent subsequence. The weak limit $0$ lies in the convex hull of $\{e_n\}$ (by Mazur's lemma) but not in the closed linear span of any finite subset — the convergence is driven by cancellation across infinitely many directions, not by approach in any particular direction.

Weak* Convergence

Definition and Relation to Weak Convergence

There is a second, closely related notion of convergence defined not on $X$ but on its dual $X^*$. The full development of the weak* topology — including the general locally convex definition, the equivalence with the Banach-space formulation, metrizability, and compactness results — is on the Weak* Topology page. Here we give the Banach-space definition and the results most relevant to weak convergence.

[definition:Weak Star Topology]
Let $X$ be a Banach space with continuous dual $X^*$. The weak* topology on $X^*$ is the coarsest topology under which every evaluation map
\begin{align*} \hat{x}: X^* &\to \mathbb{R} \\ f &\mapsto f(x) \end{align*}
is continuous, for each fixed $x \in X$.
[/definition]

As with the weak topology, the weak* topology is an initial topology — this time with respect to the family $\{\hat{x}\}_{x \in X}$ of evaluation maps. The sequential characterisation follows by the same argument as in the weak case. We write $f_n \overset{*}{\rightharpoonup} f$ to denote weak* convergence.

[quotetheorem:256]

The parallel with the Sequential Characterisation of Weak Convergence is exact: in both cases, convergence in the initial topology reduces to pointwise convergence of the defining family of maps. The only difference is which family defines the topology — $X^*$ for the weak topology on $X$, or the evaluation maps $\{\hat{x}\}_{x \in X}$ for the weak* topology on $X^*$.

The distinction between weak and weak* convergence hinges on which space acts as the test space. Both weak convergence in $X^*$ and weak* convergence in $X^*$ require convergence of real-valued sequences, but they test against different spaces:

• Weak convergence of $\{f_n\} \subseteq X^*$ tests against all elements of $X^{**}$, requiring $\Phi(f_n) \to \Phi(f)$ for every $\Phi \in X^{**}$.
• Weak* convergence of $\{f_n\} \subseteq X^*$ tests against all elements of $X$ (or more precisely, against their images $J(X) \subseteq X^{**}$), requiring $f_n(x) \to f(x)$ for every $x \in X$.

Since $J(X) \subseteq X^{**}$, every condition imposed by weak* convergence is also imposed by weak convergence. So weak convergence in $X^*$ implies weak* convergence, but not conversely — weak* convergence is the strictly weaker condition whenever $J(X) \neq X^{**}$.

Why Reflexivity Determines the Gap

When $X$ is reflexive — meaning $J: X \to X^{**}$ is surjective — the two notions coincide on $X^*$, because testing against all of $X^{**}$ is the same as testing against $J(X) = X^{**}$. This is the case for $L^p(U)$ with $1 < p < \infty$: the dual of $L^p$ is $L^q$, the bidual is $(L^q)^* \cong L^p$ again, and the canonical embedding $J$ is the identity under this identification.

When $X$ is not reflexive, the gap is real, and the most important example in PDE theory is $X = L^1(U)$. Here $X^* = L^\infty(U)$, and a bounded sequence in $L^\infty(U)$ always has a weak* convergent subsequence (testing against $L^1$ functions). But weak convergence in $L^\infty$ would require testing against all of $(L^\infty)^*$, which includes finitely additive measures and other pathological objects beyond $L^1$. The weak* topology on $L^\infty$ is therefore much coarser than the weak topology, and correspondingly much more sequences converge in it.

The Banach-Alaoglu Theorem

[quotetheorem:212]

The Banach-Alaoglu theorem is the fundamental compactness result for the weak* topology, analogous to Heine-Borel in finite dimensions. The proof embeds the unit ball of $X^*$ into a product of compact intervals $\prod_{x \in X} [-\|x\|, \|x\|]$ via the evaluation map $f \mapsto (f(x))_{x \in X}$, shows that the image is closed (linearity and the norm bound are preserved under limits), and invokes Tychonoff's theorem to conclude compactness. The result holds for all normed spaces — no separability, reflexivity, or completeness is assumed.

For reflexive $X$, the Banach-Alaoglu theorem implies the Weak Sequential Compactness theorem via the identification $X \cong X^{**}$: weak* compactness of the ball in $X^{**}$ translates directly into weak compactness of the ball in $X$. This is the mechanism by which the abstract topological compactness of Banach-Alaoglu gets converted into the sequential compactness that is directly useful in PDE arguments.

Applications to PDEs

The Galerkin Method and Weak Limits

The primary use of weak compactness in PDE theory is to extract a solution from a sequence of finite-dimensional approximations. The following example illustrates the pattern.

[example:Galerkin Weak Limit]
Let $U \subset \mathbb{R}^n$ be a bounded open domain, fix $1 < p < \infty$, and suppose $\{u_n\}_{n=1}^\infty \subseteq W_0^{1,p}(U)$ is a sequence of Galerkin approximations satisfying a uniform energy bound
\begin{align*} \|u_n\|_{W^{1,p}(U)} \leq C \quad \text{for all } n \in \mathbb{N}. \end{align*}
Since $W^{1,p}(U)$ is reflexive for $1 < p < \infty$, the Weak Sequential Compactness theorem provides a subsequence $u_{n_k}$ and a limit $u \in W_0^{1,p}(U)$ with $u_{n_k} \rightharpoonup u$ weakly in $W^{1,p}(U)$. In particular, both $u_{n_k} \rightharpoonup u$ in $L^p(U)$ and $\nabla u_{n_k} \rightharpoonup \nabla u$ in $L^p(U;\mathbb{R}^n)$. If each $u_n$ satisfies the weak formulation
\begin{align*} \int_U A \nabla u_n \cdot \nabla \phi \, d\mathcal{L}^n = \int_U f\, \phi \, d\mathcal{L}^n \quad \text{for all } \phi \in W_0^{1,p}(U), \end{align*}
with $A$ a bounded elliptic matrix and $f \in L^q(U)$, then for each fixed test function $\phi$, the map $u \mapsto \int A \nabla u \cdot \nabla \phi \, d\mathcal{L}^n$ is a continuous linear functional on $W^{1,p}$, so weak convergence gives
\begin{align*} \int_U A \nabla u \cdot \nabla \phi \, d\mathcal{L}^n = \lim_{k \to \infty} \int_U A \nabla u_{n_k} \cdot \nabla \phi \, d\mathcal{L}^n = \int_U f\, \phi \, d\mathcal{L}^n. \end{align*}
The limit $u$ therefore satisfies the same weak equation, and the energy bound passes to the limit via Lower Semicontinuity of the Norm: $\|u\|_{W^{1,p}} \leq \liminf \|u_{n_k}\|_{W^{1,p}} \leq C$.

The argument is specific to $p > 1$. For $p = 1$, the space $W^{1,1}(U)$ is not reflexive and bounded sequences need not have weakly convergent subsequences in $W^{1,1}$; they may only converge in the space of functions of bounded variation.
[/example]

Weak* Convergence in Homogenisation

In many PDE problems one works with sequences of functions that are only bounded in $L^\infty$, not in any reflexive space. Weak* compactness is then the right tool.

[example:Weak Star In Homogenisation]
Let $U \subset \mathbb{R}^n$ be a bounded domain. Consider a sequence of variable coefficients $\{a_n\}_{n=1}^\infty \subseteq L^\infty(U)$ arising from a homogenisation problem with rapidly oscillating microstructure, satisfying
\begin{align*} 0 < \lambda \leq a_n(x) \leq \Lambda < \infty \quad \text{for a.e. } x \in U \text{ and all } n \in \mathbb{N}. \end{align*}
Since $L^\infty(U) = (L^1(U))^*$ and the sequence is bounded in $L^\infty$, the Banach-Alaoglu theorem provides a subsequence $a_{n_k}$ and a limit $a \in L^\infty(U)$ with $a_{n_k} \overset{*}{\rightharpoonup} a$ weak* in $L^\infty(U)$, meaning
\begin{align*} \int_U a_{n_k}(x)\, \phi(x) \, d\mathcal{L}^n(x) \to \int_U a(x)\, \phi(x) \, d\mathcal{L}^n(x) \quad \text{for every } \phi \in L^1(U). \end{align*}
Now suppose $u_n \in H_0^1(U)$ solves $-\nabla \cdot (a_n \nabla u_n) = f$ weakly, with $f \in H^{-1}(U)$ fixed. The energy estimate $\|u_n\|_{H^1} \leq \lambda^{-1}\|f\|_{H^{-1}}$ gives a bounded sequence in $H_0^1(U)$, which is reflexive. Weak Sequential Compactness yields $u_{n_k} \rightharpoonup u$ in $H_0^1(U)$. Passing to the limit requires identifying the product $a_{n_k} \nabla u_{n_k}$, where one factor converges weak* in $L^\infty$ and the other converges weakly in $L^2$. The product of two weakly convergent sequences need not converge to the product of the limits — this is the fundamental nonlinearity obstruction. Determining the correct effective coefficient $a^*$ such that $u$ solves $-\nabla \cdot (a^* \nabla u) = f$ is precisely the problem of homogenisation, addressed by $G$-convergence and $H$-convergence theory.
[/example]

What Weak Convergence Cannot Do

The Galerkin example succeeds because the PDE is linear in $\nabla u$: the map $\nabla u \mapsto \int A \nabla u \cdot \nabla \phi \, d\mathcal{L}^n$ is a continuous linear functional, and linear functionals pass through weak limits by definition. For nonlinear problems — such as the $p$-Laplacian $-\nabla \cdot (|\nabla u|^{p-2} \nabla u) = f$ — the map $\nabla u \mapsto |\nabla u|^{p-2} \nabla u$ is not linear, and weak convergence $\nabla u_n \rightharpoonup \nabla u$ does not imply $|\nabla u_n|^{p-2} \nabla u_n \rightharpoonup |\nabla u|^{p-2} \nabla u$. Passing to the limit in nonlinear equations requires additional structure: monotonicity methods, compensated compactness, or strong convergence of the gradients obtained through improved estimates. The theory of weak convergence provides the candidate limit, but verifying that the limit solves the nonlinear equation is where the real work begins.

References

1. L.C. Evans, Partial Differential Equations (1998).
2. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (2010).
3. W. Rudin, Functional Analysis (1991).