= Solution
The <weak topology> $\sigma(X,X^*)$ is the coarsest topology on $X$ making every <continuous linear functional> $f\in X^*$ continuous. A basic neighborhood of $x_0$ is specified by finitely many inequalities $|f_j(x-x_0)|<\epsilon$.
The <weak-star topology> $\sigma(X^*,X)$ is the coarsest topology on $X^*$ making every evaluation $f\mapsto f(x)$, for $x\in X$, continuous. Its basic neighborhoods use finitely many such evaluations. Nets converge in these topologies exactly when all the corresponding <scalar> evaluations converge.
Let $J_X:X\to X^{**}$ be the canonical isometric embedding, $J_Xx(f)=f(x)$. \b[<Goldstine theorem>] states
$$
\boxed{\overline{J_X(B_X)}^{\sigma(X^{**},X^*)}=B_{X^{**}}.}
$$
Here the balls are closed <unit balls>. Thus any bidual unit <vector> can be approximated in finitely many dual evaluations by unit <vectors> from $X$; this does not claim <norm> density or sequential density in <complete> generality.
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