Product topology on $R^\omega$ and Banach Spaces

Question

Question 1: Does there exist norm on $R^\omega$ such that topology is same as product topology?

Question 2: Does there exist norm on $R^\omega$ such that corresponding normed space is Banach and topology is same as product topology?

Answer 1

You can use the fact, that $[0,1]^\omega$ is not sequentially-compact but by Tychonoff's theorem compact, to see that $\mathbb{R}^\omega$ with the product-topology is not-metrizable.

Answer 2

The product topology on $\mathbb R^\omega$ is the initial topology with respect to the projections $\pi_n$ defined by $x\mapsto x_n$. Any neighbourhood $U$ of $0$ thus contains a set of the form $\bigcap\limits_{n\in E} \pi_n^{-1}((-r_n,r_n))$ for a finite set $E$ and some $r_n>0$. Hence $U$ contains huge (co-finite dimensional) subspaces whereas balls with respect to a norm cannot contain non-trivial subspaces.