Question

1. Let V be a Nuclear space (not necessary Frechet, but complete). Let V^* be the dual space considered with the weak topology (i.e. pointwise convergence). Is it true that $V^*$ is also nuclear?

2. Is it true that the map to the double dual $V \to V^{**}$ is a linear isomorphism (not necessary continuously invertible)?

3. Let $W$ be a space like $V$. Is the map $V^* \hat \otimes W \to Hom(V,W)$ is a linear isomorphism? Here $V^* \hat{ \otimes} W $ is the completed tensor product and $Hom(V,W)$ is the space of continuous linear maps. If yes what topology does it induce on $Hom(V,W)$? If not what are the kernel and the image of this map?

4. The same questions for different kinds of duals.