Weak topologies deliberately forget norm distance and remember only scalar tests by continuous linear functionals. They are weaker but often more compact, which is why they appear throughout existence arguments.
Weak topology
The weak topology is the topology of all continuous linear observations. It keeps precisely the information visible to the dual space.
Definition. The weak topology \(\sigma(X,X^*)\) on a normed space \(X\) is the coarsest topology for which every \(f\in X^*\) is continuous. A basic neighborhood of \(x_0\) has the form \(\{x:|f_i(x-x_0)|<\varepsilon,\ i=1,\ldots,n\}\).
Proposition. A net \(x_\alpha\) converges weakly to \(x\) iff \(f(x_\alpha)\to f(x)\) for every \(f\in X^*\).
Proof. If \(x_\alpha\to x\) in the weak topology, continuity of each \(f\) gives scalar convergence. Conversely, scalar convergence for every finite list \(f_1,\ldots,f_n\) eventually puts \(x_\alpha\) in every basic weak neighborhood of \(x\), which is exactly net convergence.
Weak-star topology
The weak-star topology is pointwise convergence on the predual. It is weaker than the dual norm topology and is designed for compactness.
Definition. The weak-star topology on \(X^*\) is \(\sigma(X^*,X)\), the coarsest topology making each evaluation \(f\mapsto f(x)\) continuous for \(x\in X\). Thus \(f_\alpha\to f\) weak-star iff \(f_\alpha(x)\to f(x)\) for every \(x\in X\).
Proposition. The weak-star topology on \(X^*\) is weaker than the weak topology \(\sigma(X^*,X^{**})\). They agree on \(X^*\) only in special situations, for instance when \(X\) is reflexive.
Proof. Every evaluation by \(x\in X\) is evaluation by \(J_Xx\in X^{**}\). The weak topology on \(X^*\) asks for continuity of all members of \(X^{**}\), while the weak-star topology asks only for those coming from \(J_X(X)\). Hence weak-star convergence imposes fewer scalar tests. If \(J_X\) is onto, the two families of tests coincide.
Boundedness of weakly convergent sequences
Weak convergence cannot see norm convergence, but it still implies norm boundedness. This is another uniform boundedness consequence.
Theorem. If \(X\) is Banach and \(x_n\to x\) weakly, then \(\sup_n\|x_n\|<\infty\).
Proof. View \(x_n\) as \(J_Xx_n\in X^{**}\). For each \(f\in X^*\), the scalar sequence \(J_Xx_n(f)=f(x_n)\) converges, hence is bounded. The uniform boundedness principle applied to the family \(\{J_Xx_n:n\ge1\}\subset\mathcal L(X^*,\mathbb F)\) gives \(\sup_n\|J_Xx_n\|<\infty\). Since \(J_X\) is an isometry, \(\sup_n\|x_n\|<\infty\).
Convex closed sets
Convexity makes weak closure and norm closure agree. Separation by continuous linear functionals is the reason.
Theorem. A convex subset \(C\) of a normed space \(X\) is weakly closed iff it is norm closed.
Proof. Weakly closed implies norm closed because the weak topology is weaker than the norm topology, so weakly closed sets are closed for the stronger norm topology. Conversely, if \(C\) is norm closed and convex and \(x_0\notin C\), Hahn-Banach separation gives \(f\in X^*\) and \(\alpha\) with \(f(c)\le\alpha<f(x_0)\). The weakly open half-space \(\{x:f(x)>\alpha\}\) contains \(x_0\) and misses \(C\). Thus the complement of \(C\) is weakly open.
Standard basis example
The standard basis example shows how weak convergence can be much weaker than norm convergence in infinite dimension.
Example. For \(1<p<\infty\), the standard basis \(e_n\) in \(\ell^p\) converges weakly to \(0\). Indeed \((\ell^p)^*=\ell^q\) with \(1/p+1/q=1\), and for every \(a=(a_n)\in\ell^q\), \(a_n\to0\), so \(a(e_n)=a_n\to0\). But \(\|e_n\|_p=1\), so convergence is not in norm.
Nets and sequential caution
Weak and weak-star compactness are not always sequential. Nets are the general language needed for compactness outside metrizable settings.
Remark. Weak topologies on infinite-dimensional spaces are typically not first countable. Compactness and closure statements are therefore naturally formulated with nets or filters. On bounded subsets of duals of separable spaces, Banach-Alaoglu often supplies metrizability, and sequences become sufficient there.