Weak and Weak* Topologies - Core Definitions

Weak topologies provide alternative notions of convergence that are coarser than norm convergence but have better compactness properties, making them essential for existence proofs in functional analysis.

DefinitionWeak Topology

Let $X$ be a normed space. The weak topology on $X$ is the weakest topology making all functionals $\phi \in X^*$ continuous.

A sequence $(x_n)$ converges weakly to $x \in X$ , written $x_n \rightharpoonup x$ , if $\phi(x_n) \to \phi(x)$ for all $\phi \in X^*$ .

Weak convergence is strictly weaker than norm convergence: norm convergence implies weak convergence, but the converse fails in infinite dimensions.

ExampleWeak vs Norm Convergence

Standard Basis in $\ell^2$ : The sequence $e_n = (0, \ldots, 0, 1, 0, \ldots)$ converges weakly to $0$ (since $\phi(e_n) \to 0$ for all $\phi \in (\ell^2)^* = \ell^2$ ), but $\|e_n\| = 1$ for all $n$ , so $e_n$ does not converge to $0$ in norm
Trigonometric Functions: In $L^2[0, 2\pi]$ , the sequence $f_n(x) = \sin(nx)$ converges weakly to $0$ but not in norm
Finite Dimensions: In finite-dimensional spaces, weak convergence and norm convergence coincide

DefinitionWeak* Topology

Let $X$ be a normed space. The weak topology* on $X^*$ is the weakest topology making all evaluation maps $\phi \mapsto \phi(x)$ continuous for $x \in X$ .

A sequence $(\phi_n)$ in $X^*$ converges weak* to $\phi \in X^*$ , written $\phi_n \xrightarrow{w^*} \phi$ , if $\phi_n(x) \to \phi(x)$ for all $x \in X$ .

ExampleWeak* Convergence

Consider $X = c_0$ (sequences converging to $0$ ) with dual $X^* = \ell^1$ . The sequence $\phi_n \in \ell^1$ defined by $\phi_n = (1, 1, \ldots, 1, 0, 0, \ldots)$ (with $n$ ones) does not converge in norm, but converges weak* to $(1, 1, 1, \ldots) \notin \ell^1$ when restricted to evaluations on $c_0$ .

Actually, for $(x_k) \in c_0$ : $\phi_n((x_k)) = \sum_{k=1}^n x_k \to \sum_{k=1}^\infty x_k$ as $n \to \infty$ .

TheoremBasic Properties

Weak and weak* topologies are Hausdorff
Norm-closed convex sets are weakly closed
Weakly convergent sequences are norm bounded
In reflexive spaces, weak and weak* convergence of functionals coincide

Remark

Weak topologies sacrifice the ability to use norm estimates but gain compactness properties (via Banach-Alaoglu). This trade-off is essential for proving existence of solutions to variational problems, where minimizing sequences may not converge in norm but do converge weakly.

Weak and weak* topologies are fundamental tools in modern analysis, particularly in PDEs, optimization, and probability theory.