A reflexive Banach space with an unconditional basis admits an equivalent 1-unconditional 2 norm and embeds into a reflexive space with a 1-symmetric 2 norm. Partial results on 1-symmetric 2 renormings of spaces with a symmetric basis are obtained.

We prove that the Hermite functions are an absolute Schauder basis for many global weighted spaces of ultradifferentiable functions in the matrix weighted setting and we determine also the corresponding coefficient spaces, thus extending the previous work by Langenbruch. As a consequence, we give very general conditions for these spaces to be nuclear. In particular, we obtain the corresponding results for spaces defined by weight functions.

Supplementing and expanding classical results, for compact spaces and , metric, and their Banach spaces and of continuous real-valued functions, we provide several characterizations of the existence of isometric, resp. isomorphic, embeddings of into . In particular, we show that if the embedded space is separable, then the classical theorems of Holsztyński and Gordon become equivalences. We also obtain new results describing the relative cellularities of the perfect kernel of a given compact space and of the Cantor-Bendixson derived sets of of countable order in terms of the presence of isometric copies of specific spaces inside .