We consider the space ${}^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on ${}^{ℳ}$. A description of the space ${(}^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

This paper studies the relationship between the bidual of the (projective) tensor product of Banach spaces and the tensor product of their biduals.

We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality ||Id + P|| = 1 + ||P|| is satisfied for all weakly compact polynomials P: X → X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We also study the alternative Daugavet equation $ma{x}_{\left|\omega \right|=1}\left|\right|Id+\omega P\left|\right|=1+\left|\right|P\left|\right|$ for polynomials P: X → X. We show that this equation holds for every polynomial on the complex space X =...

An explicit representation of the n-fold symmetric tensor product (equipped with a natural topology τ such as the projective, injective or inductive one) of the finite direct sum of locally convex spaces is presented. The formula for ${⨂}_{\tau ,s}^n\left(F_1⨁F_2\right)$ gives a direct proof of a recent result of Díaz and Dineen (and generalizes it to other topologies τ) that the n-fold projective symmetric and the n-fold projective “full” tensor product of a locally convex space E are isomorphic if E is isomorphic to its square $E^2$.