Operator formulation of interacting string field theory. Pt. 1

We present an explicit, operator construction of Witten's interacting string field theory. The approach is based on a Fock space representation of string functionals and δ-function overlaps. The construction is achieved by employing conformal mapping and Neumann function techniques. The gauge invariant vertex is constructed; on physical states it is shown to lead to correct dual model amplitudes. The symmetry properties of the theory are established. (orig.)