Let $ X= \{u=(u_1, u_2, \ldots): u_n \ne 0 \text{ only for a finite number of terms}\}\subseteq\mathbb R^\mathbb N, $ with the topology inherited from $\mathbb R^\mathbb N$ (the "pointwise convergence topology"). Find the bounded sets and prove they're compact.
I show you what I've done. First of all, I think - but I'm not sure on how to prove it - the topology of $X$ is equivalent to the one generated by the (countable) family of seminorms $ p_n(u)=\vert u_n\vert, \qquad n \in \mathbb N. $ This allows me also to write an "explicit" form of an invariant metric $d$, e.g. $ d(x,y) = \sum_{n \in \mathbb N} 2^{-n}\frac{p_n(x-y)}{1+p_n(x-y)} $ Do you agree? Have you got any idea about how I can prove that the two topologies are the same? Is it true?
Anyway, let us come back to bounded sets. If my idea is correct, bounded sets of $X$ are exactly the ones for which every $p_n$ is bounded (as real valued functions). Is there a more explicit description?
And what about compactness? I suppose I have to use some theorem about compactness (Tychonov?) but I can't see how.
Thanks in advance.