We exhibit the first examples of Fréchet spaces which contain a closed infinite dimensional subspace of universal series, but no restricted universal series. We consider classical Fréchet spaces of infinitely differentiable functions which do not admit a continuous norm. Furthermore, this leads us to establish some more general results for sequences of operators acting on Fréchet spaces with or without a continuous norm. Additionally, we give a characterization of the existence of a closed subspace...

We give another version of the recently developed abstract theory of universal series to exhibit a necessary and sufficient condition of polynomial approximation type for the existence of universal elements. This certainly covers the case of simultaneous approximation with a sequence of continuous linear mappings. In the case of a sequence of unbounded operators the same condition ensures existence and density of universal elements. Several known results, stronger statements or new results can be...

We improve a result of Charpentier [Studia Math. 198 (2010)]. We prove that even on Fréchet spaces with a continuous norm, the existence of only one restrictively universal series implies the existence of a closed infinite-dimensional subspace of restrictively universal series.

A holomorphic function $f$ on a simply connected domain $\Omega$ is said to possess a universal Taylor series about a point in $\Omega$ if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta $K$ outside $\Omega$ (provided only that $K$ has connected complement). This paper shows that this property is not conformally invariant, and, in the case where $\Omega$ is the unit disc, that such functions have extreme angular boundary behaviour.