If $\mathbb{E}^*(\mathbb{R}^n)$ is the topological dual of the space o smooth function on $\mathbb{R}^n$ and $\mathbb{S}^*(\mathbb{R}^n)$ is the topological dual of the Schwartz space $\mathbb{S}(\mathbb{R}^n)$ then do we know of the any of the following is true?
1)$\mathbb{E}^*(\mathbb{R}^n)$ endowed with the weak* topology is metrizable
2) $\mathbb{S}^*(\mathbb{R}^n)$endowed with the weak* topology is metrizable
we know that $\mathbb{S}(\mathbb{R}^n)$ and $\mathbb{E}(\mathbb{R}^n)$ are Frechet spaces and separable.
I am asking because in Dieudonne's Treatise in Analysis Volume VI there is a function $F:\mathbb{S}^*(\mathbb{R}^n)\rightarrow\mathbb{D}^*(\mathbb{R}^n)$ that we know that is linear and we want to prove that is continuous and it proves it using sequences! The same to prove that a linear function $F:\mathbb{E}^*(\mathbb{R}^n)\rightarrow\mathbb{S}^*(\mathbb{R}^n)$ is continuous.
We know that convergence on the topological dual of a space X is pointwise convergence.But why continuity is equivalent with sequential continuity?
Except if in general when the topological dual of a space X is endowed with the weak* topology then the notion of continuity means sequential continuity.I mean that we use the word continuity but we actually mean sequential continuity!
