  We introduce the notion of a pseudomultiplier of a Hilbert space $\mathcal H$
of functions on a set $\Omega$. Roughly, a pseudomultiplier of $\mathcal H$ is
a function which multiplies a finite-codimensional subspace of $\mathcal H$
into $\mathcal H$, where we allow the possibility that a pseudomultiplier is
not defined on all of $\Omega$. A pseudomultiplier of $\mathcal H$ has
singularities, which comprise a subspace of $\mathcal H$, and generalize the
concept of singularities of an analytic function, even though the elements of
$\mathcal H$ need not enjoy any sort of analyticity. We analyse the natures of
these singularities, and obtain a broad classification of them in
function-theoretic terms.
