Convex-block cancellation of a weak-star limit
= Convex-block cancellation of a weak-star limit
If a bounded sequence $(y_n)$ in $X$ converges weak-star to $\phi\in X^{**}$ under the <canonical embedding into the bidual>, then any <convex blocks> $(u_n)$ have the same scalar limit on every $f\in X^*$. Thus $y_n-u_n$ is a <weakly null sequence>, even when $\phi$ does not belong to the embedded space $X$.