A sequence in a Banach space is weakly convergent to a vector if converges to for every continuous linear functional in the dual The sequence is a weakly Cauchy sequence if converges to a scalar limit for every in A sequence in the dual is weakly* convergent to a functional if converges to for every in Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the Banach–Steinhaus theorem.
A sequence in a Banach space is weakly convergent to a vector if converges to for every continuous linear functional in the dual The sequence is a weakly Cauchy sequence if converges to a scalar limit for every in A sequence in the dual is weakly* convergent to a functional if converges to for every in Weakly Cauchy sequences weakly convergent and weakly* convergent sequences are norm bounded as a consequence of the Banach–Steinhaus theorem