= Solution
The <weak topology> on a <normed vector space> $X$ is $\sigma(X,X^*)$, the coarsest <topology> making every <bounded linear functional> continuous. A neighbourhood base at $x$ is given by finitely many inequalities $|f_j(y-x)|<\varepsilon$, with $f_j\in X^*$. In the complex case, separation uses <real parts> of <bounded linear functionals>.
<Mazur theorem> states that the <weak closure> of a <convex set> equals its closure in the <norm topology>. The <weak topology> is coarser than the <norm topology>, so the norm closure is contained in the <weak closure>. Conversely, if $x$ is outside the norm closure of a <convex set> $C$, the <Hahn-Banach separation theorem> gives $f\in X^*$ and $a\in\mathbb R$ with $\operatorname{Re}f(x)>a\geq\sup_{c\in C}\operatorname{Re}f(c)$. The corresponding <weak topology> neighbourhood of $x$ misses $C$, so $x$ is outside its <weak closure>. This proves <Mazur theorem>, including the empty-set case. In particular, if $x_n$ converges weakly to $x$, then $x$ is in the <weak closure> of every tail and therefore in the norm closure of its <convex hull>. Choosing a finite <convex combination> of the $n$th tail within $1/n$ of $x$ proves the usual <Mazur lemma> formulation as well.
The <weak-star topology> on the <continuous dual space> $X^*$ is $\sigma(X^*,X)$: convergence means pointwise convergence on $X$, and a neighbourhood base prescribes finitely many evaluation inequalities. The <Banach-Alaoglu theorem> states that the <closed unit ball> of $X^*$ is compact in this <weak-star topology>, even if $X$ is incomplete. Embed this <closed unit ball> into
$$
P=\prod_{x\in X}\{z\in\mathbb K:|z|\leq\|x\|\},\qquad f\longmapsto(f(x))_{x\in X},
$$
where $\mathbb K=\mathbb R$ or $\mathbb C$. Each factor is compact, so $P$ is compact by the <Tychonoff theorem>. Inside $P$, the equations $a_{x+y}=a_x+a_y$ and $a_{\lambda x}=\lambda a_x$ define a <closed set>. Every such point defines a <linear functional> satisfying $|a_x|\leq\|x\|$, hence belongs to the <closed unit ball> of $X^*$. Thus this image is closed in $P$. The <product topology> on it is exactly the <weak-star topology>, proving <Banach-Alaoglu theorem>. Evaluations also separate its points, so the <weak-star topology> is Hausdorff.
For the <canonical embedding into the bidual> $J_X:X\to X^{**}$, we have $(J_Xx)(f)=f(x)$. Restricting all evaluations at $f\in X^*$ therefore gives exactly $\sigma(X,X^*)$. Thus \b[the induced <subspace topology> is the <weak topology> on $X$].
Now identify $X$ with $J_XX$. Let $C$ be a bounded <convex set>, write $D$ for its norm closure in $X$, and let $K$ be its <weak-star topology> closure in $X^{**}$. Boundedness places $K$ in a multiple of the <closed unit ball> of $X^{**}$, so $K$ is compact by <Banach-Alaoglu theorem>. The preceding <subspace topology> identification and <Mazur theorem> give
$$
K\cap X=\overline C^{\,w}=D.
$$
If $D$ is a <weakly compact set>, its image in the Hausdorff <weak-star topology> is compact and therefore closed. It contains $C$, so $K\subseteq D\subseteq X$. Conversely, if $K\subseteq X$, the displayed identity gives $K=D$, and its compactness is precisely weak compactness in $X$. Hence \b[$\boxed{D\text{ is weakly compact}\iff K\subseteq X}$]. This argument also covers $C=\varnothing$.
For a <bounded linear operator> $T:X\to Y$, its <Banach-space adjoint> is $T^*:Y^*\to X^*$, defined by $(T^*y^*)(x)=y^*(Tx)$. Evaluation at any fixed $x$ is thus evaluation at $Tx$ after applying $T^*$. Each is continuous in the relevant <weak-star topology>, proving that \b[$T^*$ is weak-star continuous]. Applying the same result to $T^*$ shows that $T^{**}:X^{**}\to Y^{**}$ is weak-star continuous. Direct evaluation gives
$$
T^{**}J_X=J_YT.
$$
Use $B_X$ for the <closed unit ball>; using the open ball gives the same norm closure of $T(B_X)$. <Goldstine theorem> says that $J_XB_X$ is weak-star dense in $B_{X^{**}}$. Put $K=T^{**}(B_{X^{**}})$. It is compact and closed in the <weak-star topology> by <Banach-Alaoglu theorem> and the established continuity. It contains $J_YT(B_X)$. Conversely, <Goldstine theorem> gives, for every $x^{**}\in B_{X^{**}}$, a <net> $(J_Xx_\alpha)$ from $J_XB_X$ converging weak-star to $x^{**}$; its image converges weak-star to $T^{**}x^{**}$. Consequently
$$
\overline{J_YT(B_X)}^{\,w^*}=T^{**}(B_{X^{**}}).
$$
Apply the preceding bounded <convex set> criterion in $Y$, and then scale the <closed unit ball>. We obtain the <bidual characterization of weakly compact operators>:
$$
\boxed{\overline{T(B_X)}^{\,\|\cdot\|}\text{ is weakly compact}
\iff T^{**}(X^{**})\subseteq J_YY.}
$$
The <canonical embedding into the bidual> on the right specifies exactly which copy of $Y$ is intended.
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