Could you please help me answering the following question?
Consider the Köthe space $K_ {\infty}(n^p) = \{ x= (x_n)_1^{\infty}: |x|_p := \sup_n|x_n|n^p<\infty, \forall p \in \mathbb{N} \}$ with the topology given by the norms $(|.|_p)_{p=1}^{\infty}$.
Show that $K_ {\infty}(n^p)$ is a Montel space, and the set $\{ exp(-log^{\beta}n))_{n=1}^{\infty}, 2<\beta<\infty \}$ is precompact but $\{ exp(-log^{\beta}n))_{n=1}^{\infty}, 1<\beta<2 \}$ is not.
Any hint will be appreciated.
The space $s$ of rapidely decreasing sequences is in fact a Schwartz space (it is even nuclear but this is not important for your question), i.e. the inclusions $(s,|\cdot|_{p+1}|) \hookrightarrow (s,|\cdot|_p)$ map the unit ball of the former space to a precompact set of the latter. This comes from the observation that the quotients of weights $n^p$ converge to $0$. Using this you get that bounded sets in $s$ are precompact and, since $s$ is completely metrizable, they are relatively compact.
To check precompactness of a set $B$ you just need boundedness, i.e., $\sup\lbrace |x|_p: x\in B \rbrace <\infty$ for all $p\in\mathbb N$.