I asked this a weak ago at math.stackexchange, but without success.

As far as I understand, there are several meanings of the notion of the Köthe sequence space, in particular, Hans Jarchow in his "Locally convex spaces" defines it as the space $\Lambda(P)$ of sequences $\lambda:{\mathbb N}\to {\mathbb C}$ satisfying the condition $$ \forall \alpha\in P\quad \sum_{n=1}^\infty \alpha_n\cdot|\lambda_n|<\infty, $$ 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 $\Lambda(P)$, like barreledeness, nuclearity, reflexivity, Heine-Borel property, completeness in different senses, etc. can be stated on one page (these are properties of $\Lambda(P)$ as a topological vector space, but its properties as just a vector space are interesting as well). Can anybody enlighten me if such a text exists?

Jarchow gives some conditions (for example, on p.497 he explains when $\Lambda(P)$ is nuclear), but the whole picture remains vague, and I even must confess that some elementary properties of $\Lambda(P)$ are not clear for me. For example, is it true, that if a sequence $\omega_n\ge 0$ has the property $$ \forall\lambda\in \Lambda(P)\quad \sum_{n=1}^\infty \omega_n\cdot|\lambda_n|<\infty $$ then there are $\alpha\in P$ and $C>0$ such that $$ \forall n\in{\mathbb N}\quad \omega_n\le C\cdot\alpha_n $$ ?

I can prove this only in the case when $P$ has a countable cofinal subset (excuse me my ignorance).

I asked this a week ago at math.stackexchange, but without success.

As far as I understand, there are several meanings of the notion of the Köthe sequence space, in particular, Hans Jarchow in his "Locally convex spaces" defines it as the space $\Lambda(P)$ of sequences $\lambda:{\mathbb N}\to {\mathbb C}$ satisfying the condition $$ \forall \alpha\in P\quad \sum_{n=1}^\infty \alpha_n\cdot|\lambda_n|<\infty, $$ 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 $\Lambda(P)$, like barreledeness, nuclearity, reflexivity, Heine-Borel property, completeness in different senses, etc. can be stated on one page (these are properties of $\Lambda(P)$ as a topological vector space, but its properties as just a vector space are interesting as well). Can anybody enlighten me if such a text exists?

Jarchow gives some conditions (for example, on p.497 he explains when $\Lambda(P)$ is nuclear), but the whole picture remains vague, and I even must confess that some elementary properties of $\Lambda(P)$ are not clear for me. For example, is it true, that if a sequence $\omega_n\ge 0$ has the property $$ \forall\lambda\in \Lambda(P)\quad \sum_{n=1}^\infty \omega_n\cdot|\lambda_n|<\infty $$ then there are $\alpha\in P$ and $C>0$ such that $$ \forall n\in{\mathbb N}\quad \omega_n\le C\cdot\alpha_n $$ ?

I can prove this only in the case when $P$ has a countable cofinal subset (excuse me my ignorance).