= Solution
For a <Banach space> $X$, <weak convergence> $u_k\rightharpoonup u$ means
$$
\ell(u_k)\to\ell(u)\quad\text{for every }\ell\in X^*.
$$
For its <continuous dual space> $X^*$, convergence in the <weak-star topology> means
$$
\ell_k(x)\to\ell(x)\quad\text{for every }x\in X.
$$
Weak convergence on $X^*$ tests against all of $X^{**}$, whereas weak-star convergence tests only against the canonical image of $X$. Thus weak convergence always implies weak-star convergence. If $X$ is a <reflexive Banach space>, that image is all of $X^{**}$, so \b[the weak and weak-star topologies on $X^*$ coincide]. Weak-star topology is defined relative to a chosen predual; it is not an extra intrinsic topology on an arbitrary Banach space without such an identification.
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