it is an infinite dimensional vector space over R. any vector space is not compact under the most usual topologies put on it. to see this may be you should try to prove that a finite dim vector space is not compact first of all.

then any space of lower dim is closed in a space of higher dim (under almost all of the topologies that we consider) so if the higher dim space is compact, it follows that a closed sunset will also be compact. ... maybe now you can fill in the details.