  We prove unboundedness and boundedness of the unsmeared and smeared chiral
vertex operators, respectively. We use elementary methods in bosonic Fock
space, only. Possible applications to conformal two - dimensional quantum field
theory, perturbation thereof, and to the perturbative construction of the
sine-Gordon model by the Epstein-Glaser method are discussed. From another
point of view the results of this paper can be looked at as a first step
towards a Hilbert space interpretation of vertex operator algebras.
