Vol. 43, No. 2, 1972

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Sesquilinear forms in infinite dimensions

Robert Piziak

Vol. 43 (1972), No. 2, 475–481

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.

Mathematical Subject Classification 2000
Primary: 46C10
Secondary: 15A63
Received: 24 June 1971
Published: 1 November 1972
Robert Piziak