  We study such Hilbert C*-modules over a C*-algebra $A$, that the Banach
$A$-dual module carries a natural structure of Hilbert $A$-module. In this
direction we prove that if $A$ is monotone complete, $M$ and $N$ are Hilbert
$A$-modules, $M$ is self-dual, and both $T:M\to N$ and its Banach $A$-dual
$T':N'\to M'$ have trivial kernels and cokernels then $M\cong N'$. With the
help of this result, for a monotone complete $C^*$-algebra $A$, we prove that
the index of any $A$-Fredholm operator can be calculated as the difference of
its kernel and cokernel, as in the Hilbert space case.
