5 A ug 1 99 8 A functional - analytic theory of vertex ( operator ) algebras , I

This paper is the first in a series of papers developing a functionalanalytic theory of vertex (operator) algebras and their representations. For an arbitrary Z-graded finitely-generated vertex algebra (V, Y,1) satisfying the standard grading-restriction axioms, a locally convex topological completion H of V is constructed. By the geometric interpretation of vertex (operator) algebras, there is a canonical linear map from V ⊗ V to V (the algebraic completion of V ) realizing linearly the conformal equivalence class of a genus-zero Riemann surface with analytically parametrized boundary obtained by deleting two ordered disjoint disks from the unit disk and by giving the obvious parametrizations to the boundary components. We extend such a linear map to a linear map from H⊗̃H (⊗̃ being the completed tensor product) to H, and prove the continuity of the extension. For any finitely-generated C-graded V -module (W,YW ) satisfying the standard grading-restriction axioms, the same method also gives a topological completion H of W and gives the continuous extensions from H⊗̃HW to H of the linear maps from V ⊗W to W realizing linearly the above conformal equivalence classes of the genus-zero Riemann surfaces with analytically parametrized boundaries.