= Weak topology
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The <weak topology> $\sigma(X,X^*)$ of a <normed vector space> is the coarsest topology making every continuous linear functional $f\in X^*$ continuous. A basic neighborhood of $x$ imposes finitely many tests $|f_j(y-x)|<\epsilon$. A net converges weakly exactly when all these scalar evaluations converge.
Norm convergence implies weak convergence, since $|f(x_n-x)|\leq\|f\|\|x_n-x\|$. The converse can fail: the standard unit vectors in $\ell^2$ converge weakly to zero, because every square-summable sequence has coordinates tending to zero, but their norms stay one. Weak topologies are useful because bounded families can have compactness properties unavailable in the norm topology, particularly in reflexive spaces. The tradeoff is that weak limits do not preserve every nonlinear operation.
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