On Köthe sequence spaces 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).
 A: You are very optimistic, Sergei! 
In the countable case (or, only slightly more general: if there exists a cofinal countable subset) $\Lambda(P)$ is Fréchet, and you find many results about this case, e.g., in the book Introduction to Functional Analysis of Meise and Vogt, chapter 27. But even in this case, the characterization when $\Lambda(P)$ is reflexive or Montel (=Heine-Borel-Property) is a quite difficult theorem (this is called the Dieudonné-Gomes theorem). Of course, for Fréchet spaces barrelledness is for free, but I don't know of a characterization in terms of $P$ in the uncountable case (this is related to the explicit question at the end of your post -- my guess is that this is not always true: The hypothesis means that $\omega$ defines a linear functional on $\Lambda(P)$ and the conclusion means its continuity). 
For the dual case of countable inductive limits of weighted Banach sequence spaces a lot of work has been done (e.g., by Bierstedt and others) to describe the dual again as weighted space and to characterize barrelledness in this situation. Again this is quite subtle, beyond the case of inductive limits of Banach spaces there are results of Vogt as well as Bierstedt and Bonet -- and if you really want to have a counterexample to your explicit question you should study their work. 
Other than reflexivity or the Heine-Borel property there are many locally convex properties which are directly defined in terms of the semi-norms (Schwartz or nuclearity) -- for such conditions it is no difference whether $P$ is countable or not.
A: OK, on request, the following reference as answer:
Pietsch, A.: Nuclear locally convex spaces, vol. 66 in Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, New York, Heidelberg, 1972. Translated from the second German edition. 
This should contain a lot of info on Koethe spaces. I liked it in particular because of the nice Grothendieck-Pietsch criterion on nuclearity. This is easily checked for Köthe spaces and, I guess, one of the ways to check nuclearity for many other function spaces: namely, you establish an isomorphism to a suitable Köthe space.
A: This is an addendum to the information that you already have.  Firstly, there are three sources of substantial results on Köthe spaces—unsurprisingly the first volume of his monograph (which also has useful references), Grothendieck‘s dissertation (and his textbook on topological vector spaces) and the monograph by Valdivia suggested above.  Secondly, as already mentioned, they are a useful source of models for spaces of test functions and distributions.  There is a useful unified approach to this which is, I hope, worth mentioning.  Suppose that we have an unbounded, self-adjoint operator $T$ on Hilbert space.  (Typically, one uses differential operators of mathematical physics whose spectral properties are well known—Sturm Liouville operators, the Laplace Beltrami operator on suitable domains or manifolds, possibly with boundary conditions, and Schrödinger operators).
Then the intersection  of the domains of definition of  its powers has a natural Fréchet space structure and many of the classical spaces of test functions can be obtained in this way.  A dual construction leads to the corresponding spaces of distributions.  The connection with Köthe spaces is provided by the fact that if the spectrum of $T$ is discrete, i.e., consists of a sequence of eigenvalues, then this space (and its dual) are Köthe spaces which can be described explicitly in terms of this sequence.  An advantage of this correspondence is that it can easily be generalised to give, e.g., ultradistributions, Roumieu spaces and Sobolev spaces, even of infinite order.
One example of the benefits is that this approach  can be used to give transparent proofs of important results in various contexts—the classical case is the celebrated kernel theorem of Laurent Schwartz.
A: Good question. I think these sequence spaces deserve to be better known because they provide a rich bank of concrete examples for things related to the theory of topological vector spaces which can be dauntingly abstract.
Another resource with an extensive discussion of these spaces is the book by Manuel Valdivia "Topics in Locally Convex Spaces". It has a long chapter on sequence spaces including the particular case of echelon spaces which was a key class of examples used in the work of Grothendieck when he discovered the notion of nuclear spaces.
By the way, my previous somewhat related question 
The "Spaces of Schwartz distributions are finite dimensional" challenge
was about figuring out nice properties of $P$ which would ensure $\Lambda(P)$ would behave, for all practical purposes, like a finite-dimensional space, i.e., it would be nuclear, (strongly) reflexive,...(fill in the blanks).
Addendum: Following Jochen's comment, I should add that providing examples is not the only motivation for spending time learning what a sequence space is. Spaces that matter are sequence spaces (modulo TVS isomorphism). I in fact would go further in this philosophy, in particular with regards to teaching distributions TVS's etc., not per se but for the needs of mathematical physics, probability,... As can be seen from my other posts listed below. Even in an introductory course, I think it makes sense investing time at the beginning to prove the sequence space isomorphism theorems once and for all, and then proving all the needed theorems like the kernel, Fubini for distributions, Bochner-Minlos, Prokhorov, Lévy continuity,...with sequence spaces.
Can distribution theory be developed Riemann-free?
https://math.stackexchange.com/questions/3512357/understanding-the-proof-of-schwartz-kernel-theorem/3512932#3512932
https://math.stackexchange.com/questions/2623515/schwartz-kernel-theorem-and-dual-topologies/2647815#2647815
