Given a vector bundle $E \to X$ over a complex curve $X$, let its rank and degree be $r$ and $d$ respectively. Then the slope of $E$ is defined to be
$ \mu(E):= \frac{d}{r}. $
$E$ is defined to be stable if $\mu(U) < \mu(E)$ for every non-trivial proper subbundle $U$ of $E$.
I'm trying to decide whether the following is true: $E$ is stable if and only if its dual bundle $E^*$ is stable.
If would like the following to hold: $\mu(U) < E \iff \mu(E/U) > \mu(E)$ because then given a subbundle $U$ I can dualize the ses
$0 \to U \to E \to E/U \to 0$
to give ses
$0 \to (E/U)^* \to E^* \to U^* \to 0$,
from which it follows that
$E^*$ stable $\implies \mu((E/U)^*) < \mu(E^*) \implies \mu((E/U))>\mu(E) \implies \mu(U) < \mu(E).$
However, this does not seem like it would hold in general, but I can't think of a counter example (i.e stable $E$ with $E^*$ not stable).
Thanks.