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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 TVSs 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, Levy 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

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

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).

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 TVSs 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, Levy 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

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.

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).