On the spaces of John Horvلth

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