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 <a href="https://books.google.com/books?id=DSwo-8sMehUC&printsec=frontcover#v=onepage&q&f=false">"Topics in Locally Convex Spaces"</a>. 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 
https://mathoverflow.net/questions/297196/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).