Weak convergence ($x_n\rightharpoonup x$)

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 $X^{\ast }$. 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.

The displayed spikes have $L^p$ norm one. Pairing with $g\in L^{p^{\prime }}$ and applying Holder inequality bounds the pairing by $({\int }_0^{1/j}|g{|}^{p^{\prime }}{)}^{1/p^{\prime }}$, which tends to zero. At exponent one, however, $j{1}_{(0,1/j)}$ cannot converge weakly in $L^1$: every interval away from zero forces any limit to vanish there, while the constant test function preserves integral one. Their failure of uniform integrability records exactly the obstruction in the Dunford-Pettis theorem.

The weak limit of a sequence in a normed vector space is an element at which every continuous linear functional has the limit of its values along the sequence. Thus the definition is precisely weak convergence to that element.

The nonnegative Lp space functions $f_j$ have $L^1$ norm one. Any weak limit tested against indicators supported outside the origin must vanish almost everywhere. Testing against the constant bounded function one would nevertheless require integral one. This contradiction applies to every subnet whose indices tend to infinity, so the $L^1$ unit ball is not weakly compact. The abstract Banach-Alaoglu theorem remains valid: weak compactness of the $L^1$ unit ball is a different assertion from weak-star compactness of a dual ball.

A normed vector space has the Radon-Riesz property when weak convergence $u_k\rightharpoonup u$ together with $\Vert u_k\Vert \to \Vert u\Vert$ implies strong convergence $u_k\to u$. The Hilbert-space case is the Radon-Riesz theorem. The condition prevents norm from remaining in weakly invisible directions.