= Weak convergence
{title2=$x_n\rightharpoonup x$}
= Weakly convergent
{synonym}
A sequence $x_n$ in a normed vector space converges weakly to $x$ when $f(x_n)\to f(x)$ for every member of the <continuous dual space> $X^*$. The notation is $x_n\rightharpoonup x$. The testing functionals are fixed while the index varies.
For $1<p<\infty$, the standard unit vectors in $\ell^p$ are weakly null: pairing with any dual vector in $\ell^q$ selects a coordinate tending to zero. Their norms nevertheless stay one. Thus weak convergence controls every fixed linear measurement without requiring norm-small error. A weakly convergent sequence is norm bounded, and adding suitable norm information can upgrade it to strong convergence in spaces with the <Radon-Riesz property>.
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