  A mapping $f: {\mathcal M} \to {\mathcal N}$ between Hilbert $C^*$-modules
approximately preserves the inner product if \[\|< f(x), f(y)> - < x, y> \|
\leq \phi(x, y),\] for an appropriate control function $\phi(x,y)$ and all $x,
y \in {\mathcal M}$. In this paper, we extend some results concerning the
stability of the orthogonality equation to the framework of Hilbert
$C^*$-modules on more general restricted domains. In particular, we investigate
some asymptotic behavior and the Hyers--Ulam--Rassias stability of the
orthogonality equation.

Operator Algebras;Analysis of PDEs;Functional Analysis