We consider the multiplicative algebra P(𝒢₊') of continuous scalar polynomials on the space 𝒢₊' of Roumieu ultradistributions on [0,∞) as well as its strong dual P'(𝒢₊'). The algebra P(𝒢₊') is densely embedded into P'(𝒢₊') and the operation of multiplication possesses a unique extension to P'(𝒢₊'), that is, P'(𝒢₊') is also an algebra. The operation of differentiation on these algebras is investigated. The polynomially extended Laplace transformation and its connections with the differentiation...

For $U$ a balanced open subset of a Fréchet space $E$ and $F$ a dual-Banach space we introduce the topology ${\tau }_{\gamma }$ on the space $\mathscr{H}(U,F)$ of holomorphic functions from $U$ into $F$. This topology allows us to construct a predual for $(\mathscr{H}(U,F),{\tau }_{\delta })$ which in turn allows us to investigate the topological structure of spaces of vector-valued holomorphic functions. In particular, we are able to give necessary and sufficient conditions for the equivalence and compatibility of various topologies on spaces of vector-valued holomorphic functions....