Given sequences $\displaystyle (a_1, a_2, ...), (b_1, b_2, ...)$ of real numbers with $\displaystyle a_i > 0$ for all $\displaystyle i$, define $\displaystyle h: \mathbb{R}^\omega \rightarrow \mathbb{R}^\omega$ by the equation $\displaystyle h((x_1, x_2,...))=(a_1x_1+b_1, a_2x_2+b_2,...)$.

1. Show that if $\displaystyle \mathbb{R}^\omega$ is given the product toopology, $\displaystyle h$ is a homeomorphism.

2. What happens if $\displaystyle \mathbb{R}^\omega$ is given the box topology?

3. If $\displaystyle \mathbb{R}^\omega$ is given the uniform topology, under what conditions on the numbers $\displaystyle a_i, b_i$ is $\displaystyle h$ continuous? a homeomorphism?

For part 1, I know how to show $\displaystyle h$ is bijective, but having trouble showing $\displaystyle h$ and $\displaystyle h^{-1}$ continuous. I don't know how to do 2 and 3. Can I get some help please?