Weak and Weak\* Convergence

6.1 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 $y \in H$.

6.2 Weak* Convergence

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.

6.3 Weak Convergence in Specific Spaces

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 $y \in \ell^2$.