This paper is concerned with
sesquilinear forms defined on vector spaces of arbitrary dimension. Motivation is
taken from classical Hilbert space theory, as the orthogonality relation induced by the
form is used to replace the topology. First, an algebraic version of the Frechet-Riesz
Representation Theorem is proved for linear functionals having an orthogonally
closed kernel. Next, the notion of adjoint is formulated, following von Neumann, in
the language of linear relations. It is proved that the adjoint of an arbitrary relation
is a single valued linear relation precisely when the domain of that relation is
orthogonally dense. Finally, an algebraic version of a continuous linear operator is
introduced and the relationship with the notion of adjoint and linear functional is
studied. The main result here is that an operator is orthogonally continuous
precisely when it has an everywhere defined adjoint. These general results of
pure algebra imply standard topological results in the context of a Hilbert
space.
