Schur property

Let $X$ be a real or complex Banach space. It has the Schur property if, for every sequence $(x_n{)}_{n\in \mathbb{N}}$ in $X$ and every $x\in X$,

$x_n⟶x\text{ in }\sigma (X,X^{\ast })⟹\mathrm{\parallel }{x}_n-x\mathrm{\parallel }⟶0.$

Thus the premise is convergence in the weak topology from Weak topology on a normed space, while the conclusion is convergence for the given norm. Equivalently, it is enough to test weakly null sequences: if $x_n\to x$ weakly, linearity of every $f\in X^{\ast }$ gives $f(x_n-x)\to 0$; conversely this applied to $x_n-x$ recovers weak convergence to $x$. The corresponding norm statements are equivalent because $\mathrm{\parallel }(x_n-x)-0\mathrm{\parallel }=\mathrm{\parallel }{x}_n-x\mathrm{\parallel }$.

The definition concerns sequences only. It does not say that the weak and norm topologies coincide, nor does it turn weak convergence of arbitrary nets into norm convergence. The zero Banach space has the Schur property.