 Weak convergence

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.