A functional-analytic theory of vertex (operator) algebras, I
  This paper is the first in a series of papers developing a functional-analytic 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 the tensor product of V and V to 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 the completed tensor product of H and H to H, and prove the continuity of the extension. For any finitely-generated C-graded V-module (W, Y_W) satisfying the standard grading-restriction axioms, the same method also gives a topological completion H^W of W and gives the continuous extensions from the completed tensor product of H and H^W to H^W of the linear maps from the tensor product of V and W to the algenbraic completion of W realizing linearly the above conformal equivalence classes of the genus-zero Riemann surfaces with analytically parametrized boundaries. 
Yi-Zhi Huang