= Solution
Let $K=\overline{T(B_X)}$, which is compact. Every $y^*$ in the unit ball of the <continuous dual space> $Y^*$ restricts to a function on $K$. These functions are uniformly bounded and equicontinuous, since
$$
|y^*(z)-y^*(w)|\leq\lVert z-w\rVert.
$$
By the <Arzelà-Ascoli theorem>, every sequence $(y_n^*)$ in the unit ball has a subsequence that converges uniformly on $K$. Along this subsequence,
$$
\lVert T^*y_n^*-T^*y_m^*\rVert
=\sup_{x\in B_X}|(y_n^*-y_m^*)(Tx)|
\leq\sup_{z\in K}|(y_n^*-y_m^*)(z)|\longrightarrow0.
$$
The <continuous dual space> $X^*$ is complete, so $(T^*y_n^*)$ converges in norm. Every sequence in the image of the unit ball therefore has a convergent subsequence, proving the forward direction of the <Schauder theorem for compact operators>:
$$
\boxed{T^*:Y^*\to X^*\text{ is compact}.}
$$
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