  In this work we describe the mathematical foundations used in the
construction of primary fields of minimal models of conformal field theory. The
work contains two parts: In the first part we give a description of Verma and
Fock modules for the Virasoro algebra and develop their imbedding patterns.
This part is a simplification of the work of Feigin and Fuks (we correct a
mistake in their patterns in the case III_+), Rocha-Charidi and some new ideas
which yield a simplification of the original papers. In the second part we
define (free) vertex operators as unbounded Hilbert space operators, acting on
Fock spaces, which are Virasoro modules. We prove several properties of these
operators: under appropriate conditions vertex operators are densely defined,
not closable operators. Radially ordered products of vertex operators exists on
a dense subset. We prove commutation relations between vertex operators and
elements of the Virasoro algebra. Next we define, following the (non rigouros)
work of G. Felder, integrated vertex operators and prove that these operators
resemble the properties of the not integrated vertex operators. Special
integrated vertex operators can be identified with conformal fields and a
Virasoro invariant subspace of Fock space can be identified with the physical
Hilbert space for the conformal theory.
