real analysis: Schwarz's space is Frechet's space

I have a question about the Schwartz space integrity test in Folland, proposition 8.2.

Drink $ (f_k) $ be a Cauchy sequence in Schwartz's space $ S $.

I understand that in the test you built $ g_0 $ that satisfies

$$ partial ^ alpha f_k to partial ^ alpha g_0 $$

evenly But in the definition of the norm $ | dot | _ (N, α) $there is a factor $ (1+ | x |) ^ N $ and after taking the sup, how can we guarantee the uniform convergence of

$$ (1+ | x |) ^ N ( partial ^ alpha f_k) a (1+ | x |) ^ N ( partial ^ alpha g_0)? $$

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Functional analysis. What is the connection between Frechet's Lie groups and Lie algebras?

An ordinary Lie group has a differentiable structure, that is, it is locally isomorphic to a Euclidean space of finite dimension. A Frechet Lie group, on the other hand, has a multiple Frechet structure, that is, it is locally isomorphic to a Frechet space of infinite dimension.

My question is, what is the connection between the Lie Frechet groups and the Lie algebras? Wikipedia says this:

Some of the relationships between Lie algebras and Lie groups are still valid in this context.

But which ones are still valid and which are not?