Hans Jarchow in his "Locally convex spaces" defines the Köthe sequence space $\Lambda(P)$ by the condition $$ \lambda\in\Lambda(P)\quad\Leftrightarrow\quad \forall \alpha\in P\quad \sum_{n=1}^\infty \alpha_n\cdot|\lambda_n|<\infty, $$ where $P$ is an arbitrary set of sequences with the properties:
1) $\forall\alpha\in P$ $\forall n\in{\mathbb N}$ $\alpha_n\ge 0$,
2) $\forall\alpha,\beta\in P$ $\exists\gamma\in P$ $\forall n\in{\mathbb N}$ $\max\{\alpha_n,\beta_n\}\le\gamma_n$
3) $\forall n\in{\mathbb N}$ $\exists\alpha\in P$ $\alpha_n>0$.
Jarchow mentions the space $\Lambda(P)$ from time to time in his book to illustrate (sometimes to formulate) different results, but without a summary about $\Lambda(P)$.
I wonder if there is a text where the results on $\Lambda(P)$ are systematized? I think the main properties of $\Lambda(P)$ as a topological vector space, like barreledeness, nuclearity, reflexivity, Heine-Borel property, completeness in different senses, etc. can be stated on one page. Can anybody enlighten me if such a text exists?
P.S. Actually, the other properties of $\Lambda(P)$, where it is not considered as a topological vector space are interesting as well... Is it true, for example, that if a sequence $\{\omega_n\}$ has the property $$ \forall\lambda\in \Lambda(P)\quad \text{the series}\ \sum_{n=1}^\infty \omega_n\cdot\lambda_n \ \text{converges} $$ then there are $\alpha\in P$ and $C>0$ such that $$ \forall n\in{\mathbb N}\quad |\omega_n|\le C\cdot\alpha_n $$ ?
Edit 22.05.2020. I asked this also at MathOveflow.