I was trying to prove this result and this is my attempt.
Theorem [Schauder] Let $E,F$ be Banach spaces, and let $T:E\to F$ a compact operator. Then its adjoint $T^*:F^*\to E^*$ is compact.
Proof attempt. An operator between Banach spaces is compact iff the image of any bounded sequence has a Cauchy subsequence. So let $\left\{ v_n\right\}\subset F^*$ be a bounded sequence, with $\|v_n\|\leq C$. By the sequential Banach-Alaoglu's theorem, $\left\{ v_n\right\}$ has a weakly-star converging subsequence to some $\bar{v}\in F^*$, which we keep calling $\left\{ v_n\right\}$. Thus $$|\left\langle v_{n}-\bar{v},y\right\rangle| \to 0,\qquad \forall y\in F $$ To prove the thesis, we need to show that $\left\{ A^*v_n\right\}$ has a Cauchy subsequence, which we keep calling $\left\{ A^*v_n\right\}$. Let $B_E$ be the closed unit ball of $E$; we have \begin{align*}\|A^*v_n-A^*v_m\|_{E^*}&=\sup_{x\in B_E}|\left\langle A^*v_n-A^*v_m,x\right\rangle|=\sup_{x\in B_E}|\left\langle v_n-v_m,Ax\right\rangle|=\\ &=\sup_{y\in T(B_E)}|\left\langle v_n-v_m,y\right\rangle| \end{align*} so it suffices to show that the latter term converges to $0$ as $n,m\to +\infty$.\
To this purpose, since $T$ is compact, then $B':=\overline{T(B_E)}$ is compact in $F$. Thus, for all $n\in\mathbb{N}$ there is $y_n\in B'$ such that $$\sup_{y\in T(E)}|\left\langle v_n-\bar{v},y\right\rangle|\leq \sup_{y\in B}|\left\langle v_n-\bar{v},y\right\rangle|= |\left\langle v_n-\bar{v},y_n\right\rangle|$$ Since $\left\{y_n\right\}\subset B'$ and $B'$ is compact, $\left\{y_n\right\}$ has a converging subsequence, which we keep calling $\left\{ y_n\right\}$, so that $y_n\to \bar{y}\in B'$. Thus \begin{align*}|\left\langle v_n-\bar{v},y_n\right\rangle|&\leq |\left\langle v_n-\bar{v},y_n-\bar{y}\right\rangle|+|\left\langle v_n-\bar{v},\bar{y}\right\rangle|\leq\\ &\leq 2C\|y_n-\bar{y}\|+ |\left\langle v_n-\bar{v},\bar{y}\right\rangle|\to 0 \end{align*}
And finally $$\sup_{y\in T(E)}|\left\langle v_n-v_m,y\right\rangle|\leq \sup_{y\in T(E)}|\left\langle v_n-\bar{v},y\right\rangle|+ \sup_{y\in T(E)}|\left\langle \bar{v}-v_m,y\right\rangle|\to 0$$ and the thesis is proved.
The problem is that the sequential Banach Alaoglu's theorem holds only when $E$ is separable. Is there a way I could adapt this proof to the general case? Maybe using nets instead of sequences?
Verify that $\{v_n\}$ is equi-continuous and uniformly bounded on $C(B')$ and apply Arzela -Ascoli Theorem to get a uniformly convergent subsequence.