0.A functor F gives a map from Hom(V,W) to Hom(F(V),F(W)). Call F continuous if this map is always continuous [using the metric in the other question I asked]. Show that if $$\xi=\pi :E \to B $$is any vector bundle, and F is continuous, then there is a bundle $$F(\xi)= \pi':E'\to B$$ for which $$\pi'^{-1}(p) = F(\pi^{-1}(p))$$, and such that to every trivialization $$t: \pi^{-1} \to U \times R^n$$ corresponds a trivialization $$t' : \pi'^{-1}(U) \to U \times F(R^n).$$ 1. The functor V** is continuous.
2.the functor F(V)=V* is continuous.
3.Define a continuous contravariant functor F, and show how to construct a bundle $$F(\xi)$$