  Let $\mathbb F$ be a finite field. Consider a direct sum $V$ of an infinite
number of copies of $\mathbb F$, consider the dual space $V^\diamond$, i.~e.,
the direct product of an infinite number of copies of $\mathbb F$. Consider the
direct sum ${\mathbb V}=V\oplus V^\diamond$. The object of the paper is the
group $\mathbf{GL}$ of continuous linear operators in $\mathbb V$. We reduce
the theory of unitary representations of $\mathbf{GL}$ to projective
representations of a certain category whose morphisms are linear relations in
finite-dimensional linear spaces over $\mathbb F$. In fact we consider a
certain family $ Q_\alpha$ of subgroups in $\mathbb V$ preserving two-element
flags, show that there is a natural multiplication on spaces of double cosets
with respect to $ Q_\alpha$, and reduce this multiplication to products of
linear relations. We show that this group has type $\mathrm{I}$ and obtain an
'upper estimate' of the set of all irreducible unitary representations of
$\mathbf{GL}$.
