On the spaces OCm of John Horváth

In the table (on p. 442) of John Horváth's textbook “Topological Vector Spaces and Distributions”, the questions for an explicit description of the topology of OCm by seminorms, for the completeness and for the metrizability of the space OCm remained open, whereas, for m=∞, the completeness was shown by A. Grothendieck in his thesis. We resolve these questions in Proposition 1, Proposition 2, Proposition 3. In Proposition 4, Proposition 5, Proposition 6, Proposition 7, we investigate multiplication and convolution in the spaces Sm, OCm, S′m, OC′m analogously to the treatment in the spaces Em, Dm, D′m, E′m in J. Horváth's book (Ch. 4, §6 Multiplication, §9 Convolution, §10 Regularization). In Section 6, the multiplication and convolution of holomorphic functions with values in Sm, OCm, S′m, OC′m are investigated analogously to J. Horváth's treatment of holomorphic functions with values in the spaces Em, Dm, D′m, E′m.