1 Let H(U) denote the algebra of holomorphic functions on an open sub- set U of a complex locally convex Hausdorf space E and let H,(U) and H,(U) denote this algebra when supplied with the continuous and the associated equable convergence structures, respectively [3,4]. One objec- tive of this note is to show that the convergence algebras H,(U) and H,(U) in some (quite general) cases contain total information concerning the space U in the sense that the spectrum of H,(U) or H,(U) is homeo- morphic to U, when it is given the continuous convergence structure. The finest locally convex topologies on H(U) coarser than the con- vergence structures of H,(U) and H,(U) will be denoted by K and A. In [lS] H. Jarchow treats analogously defined topologies y and q on spaces of continuous linear forms. In [S] we proved that H,(U) coincides with H,,(U) (cf. [19]) if U is a Lindelof space and that il is always liner than the ported topology z, (cf. [9]). Isidoro [ 141 and Mujica [ 181 have characterized the spectrum (as a set) of H,,(U), Hro( U), and H,,(U) and it has been used, e.g., for constructing the envelope of holomorphy of U. Using the result about the spectrum of H,(U) and H,(U), mentioned above, we obtain information on the spectrum of H,(U) and H,(U). Finally we prove that the bornological topology associated with the con- tinuous convergence structure on the vector space H( U, F), of holomorphic functions U + F, coincides with the compact-open topology, when U is an open subset of a DFM-space E and F is a metrizable locally convex space. This result is analogous to a similar result of B. Miiller [19] concerning spaces of continuous linear functions. For topological spaces X and Y let C,(X, Y) be the set of continuous functions X + Y endowed with the continuous convergence structure, i.e., the coarsest convergence structure for which w: C(X, Y) x X-+ Y, o(f, x) =f(x) is continuous. If E is a locally convex space, then C,(X, E) is 207