  Let $B(\Omega)$ be the Banach space of holomorphic functions on a bounded
connected domain $\Omega$ in $\mathbb C^n$, which contains the ring of
polynomials on $\Omega $. In this paper, we first establish a criterion for
$B(\Omega )$ to be reflexive via evaluation functions on $B(\Omega )$, that is,
$B(\Omega )$ is reflexive if and only if the evaluation functions span the dual
spaces $(B(\Omega ))^{*} $. Moreover, under suitable assumptions on $\Omega$
and $B(\Omega)$, we establish a characterization of the composition operator
$C_\varphi$ to be a Fredholm operator on $B(\Omega)$ via the property of the
holomorphic self-map $\varphi:\Omega\to\Omega$. Our new approach utilizes the
symbols of composition operators to construct a linearly independent function
sequence, which bypasses the use of boundary behavior of reproducing kernels as
those may not be applicable in our general setting.
