Weak and Weak\* Convergence

A sequence $\{x_n\}$ in a normed space $X$ converges weakly to $x$ (written $x_n \rightharpoonup x$) if $\varphi(x_n) \to \varphi(x)$ for every $\varphi \in X^*$.

Proposition 6.1. If $x_n \to x$ in norm, then $x_n \rightharpoonup x$ (strong convergence implies weak convergence).

Proposition 6.2. If $x_n \rightharpoonup x$ is weakly convergent, then $\sup_n \|x_n\| < \infty$.

Theorem 6.3. In a Hilbert space $H$, $x_n \rightharpoonup x$ if and only if $\langle x_n, y\rangle \to \langle x, y\rangle$ for every $y \in H$.

A sequence $\{\varphi_n\} \subseteq X^*$ converges weak* to $\varphi$ (written $\varphi_n \overset{w^*}{\to} \varphi$) if $\varphi_n(x) \to \varphi(x)$ for every $x \in X$.

Theorem 6.4 (Banach-Alaoglu). The closed unit ball of $X^*$ is weak*-compact.

Theorem 6.5. In $\ell^p$ ($1 < p < \infty$), $x_n \rightharpoonup x$ if and only if $x_n$ is bounded and $x_n(i) \to x(i)$ for each coordinate $i$.

Example. In $\ell^2$, the standard basis vectors $e_n$ converge weakly to $0$ but not in norm: $\|e_n\| = 1$ for all $n$, but $\langle e_n, y\rangle = y_n \to 0$ for every $y \in \ell^2$.

Proposition 6.6 (Weak vs. Weak*). In a normed space $X$:

• If $X$ is reflexive, then weak and weak* convergence on $X^*$ coincide.
• In general, weak convergence on $X^*$ implies weak* convergence, but the converse fails.

Proposition 6.7 (Uniqueness of Limits). Weak limits and weak* limits are unique when they exist.

Proposition 6.8 (Weak Convergence in $L^p$). For $1 \leq p < \infty$, a sequence $f_n \rightharpoonup f$ in $L^p(\mu)$ if and only if $\int f_n g\, d\mu \to \int f g\, d\mu$ for every $g \in L^q(\mu)$, where $1/p + 1/q = 1$. For $p = \infty$, weak* convergence is often more useful: $f_n \overset{w^*}{\to} f$ in $L^\infty$ if $\int f_n g \to \int f g$ for every $g \in L^1$.

Lemma 6.9 (Mazur). Let $X$ be a normed space and $x_n \rightharpoonup x$. Then there exists a sequence of convex combinations $y_n = \sum_{k=n}^{N_n} \lambda_k x_k$ (with $\lambda_k \geq 0$, $\sum \lambda_k = 1$) such that $y_n \to x$ in norm.

Corollary 6.10. If $C \subseteq X$ is convex, then $C$ is weakly closed if and only if it is strongly closed.

Corollary 6.11. If $x_n \rightharpoonup x$, then $\|x\| \leq \liminf_{n\to\infty} \|x_n\|$. This is the weak lower semicontinuity of the norm.

Theorem 6.12 (Eberlein-Smulian). In a Banach space, a set is weakly compact if and only if it is weakly sequentially compact. That is, $A \subseteq X$ is weakly compact iff every sequence in $A$ has a weakly convergent subsequence with limit in $A$.

Theorem 6.13 (Kakutani). A Banach space $X$ is reflexive if and only if the closed unit ball $B_X$ is weakly compact (equivalently, weakly sequentially compact).

Corollary 6.14. In a reflexive Banach space, every bounded sequence has a weakly convergent subsequence.

Example. $L^p(\mu)$ for $1 < p < \infty$ is reflexive, so every bounded sequence in $L^p$ has a weakly convergent subsequence. $L^1(\mu)$ is not reflexive: the sequence $f_n = n\chi_{[0,1/n]}$ on $[0,1]$ is bounded in $L^1$ but has no weakly convergent subsequence.

Proposition 6.15. If $T : X \to Y$ is a bounded linear operator and $x_n \rightharpoonup x$ in $X$, then $T x_n \rightharpoonup T x$ in $Y$. That is, bounded linear operators are weakly continuous.

Proposition 6.16. If $T : X \to Y$ is a compact operator and $x_n \rightharpoonup x$ in $X$, then $T x_n \to T x$ in norm. Compact operators map weakly convergent sequences to strongly convergent sequences.

Example. In $L^2([0,1])$, the integral operator $(Tf)(x) = \int_0^1 K(x, y) f(y)\, dy$ with $K \in L^2([0,1]^2)$ is compact. If $f_n \rightharpoonup f$, then $T f_n \to T f$ in $L^2$.

Application 1: Calculus of Variations. Weak convergence is central to the direct method in the calculus of variations. To minimize a functional $I$ over a space $X$, one takes a minimizing sequence $x_n$ with $I(x_n) \to \inf I$. If $I$ is weakly lower semicontinuous and the sequence is bounded, weak compactness gives a convergent subsequence whose limit is the minimizer.

Application 2: PDE Theory. Weak solutions of PDEs are often obtained by constructing approximate solutions and extracting a weakly convergent subsequence. The existence theory for elliptic PDEs via the Lax-Milgram theorem relies on weak convergence in Hilbert spaces.

Application 3: Ergodic Theory. Von Neumann’s mean ergodic theorem states that if $T$ is a unitary operator on a Hilbert space $H$, then $\frac{1}{n}\sum_{k=0}^{n-1} T^k x$ converges weakly to the projection of $x$ onto the subspace of $T$-invariant vectors.

Problem 1. Show that $e_n \rightharpoonup 0$ in $\ell^2$ but not in $\ell^1$.

Solution. For $\ell^2$: $\langle e_n, y\rangle = y_n \to 0$ for any $y \in \ell^2$ since $\sum y_n^2 < \infty$ implies $y_n \to 0$. For $\ell^1$: $\ell^1$ has dual $\ell^\infty$. Take $\varphi \in (\ell^1)^*$ corresponding to $(1, 1, 1, \ldots) \in \ell^\infty$. Then $\varphi(e_n) = 1 \not\to 0$, so $e_n$ does not converge weakly to $0$ in $\ell^1$. $\blacksquare$

Problem 2. Let $f_n(x) = \sin(nx)$ in $L^2([0, 2\pi])$. Show $f_n \rightharpoonup 0$.

Solution. For any $g \in L^2$, the Riemann-Lebesgue lemma gives $\int_0^{2\pi} \sin(nx) g(x)\, dx \to 0$. Hence $\langle f_n, g\rangle \to 0$, so $f_n \rightharpoonup 0$. Note that $\|f_n\|_2 = \sqrt{\pi}$, so $f_n$ does not converge strongly. $\blacksquare$

1. Prove that weak limits are unique.
2. Show that if $x_n \rightharpoonup x$ and $\varphi \in X^*$, then $\varphi(x_n) \to \varphi(x)$.
3. Prove that in a finite-dimensional space, weak convergence is equivalent to norm convergence.
4. Show that $L^\infty([0,1])$ is not reflexive by finding a bounded sequence with no weakly convergent subsequence.
5. Let $T : X \to Y$ be compact. Prove that if $x_n \rightharpoonup 0$, then $T x_n \to 0$.