Weak convergence ($x_n\rightharpoonup x$)

A sequence $x_n$ converges weakly to $x$ when $f(x_n)\to f(x)$ for every $f$ in the continuous dual space $X^{\ast }$.

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.