Talk:Smith space

Examples please

User:Eozhik For those of us new to this topic, it would be useful to have some simple examples. For example, is there an elementary characterization of the stereotype duals of the classicial l_p sequence spaces? Of L^p function spaces? Kenneth.harris (talk) 16:33, 22 February 2020 (UTC) — Preceding unsigned comment added by Kenneth.harris (talk • contribs) 16:29, 22 February 2020 (UTC)

• Kenneth.harris if you mean characterizations of $(\ell _{p})^{\star }$ and $(L_{p})^{\star }$ as objects in categories of locally convex spaces or Smith spaces, as far as I know, nobody has yet been interested in this issue. The examples of Smith spaces are quite unusual, maybe they are worthy of mentioning. Eozhik (talk) 19:26, 22 February 2020 (UTC)
• I gave some simple examples. The concrete examples of Smith spaces that I know (apart from ${\mathcal {C}}^{\star }(M)$) are quite technical, I’m not sure that it will be appropriate to mention them in the encyclopedia. Eozhik (talk) 20:40, 22 February 2020 (UTC)

• Eozhik: Yes, that is what I meant. When trying to understand this subject, the first question I wondered was: if the stereotype dual of every Banach space is a Smith space, then what is the stereotype dual of separable Hilbert space, $(\ell _{2})^{\star }$? Is it homeomorphic to a space I am already familiar with? If no-one has been interested in this then OK, but as an outsider I would have thought this was an early question to ask.
• Thanks, those examples are very helpful! Kenneth.harris (talk) 14:53, 23 February 2020 (UTC)

• Kenneth, yes, I did not see any descriptions of the stereotype dual $(\ell _{2})^{\star }$. The theory of topological vector spaces was popular some time ago, and perhaps among different spaces studied in its different branches, there are examples that coincide with $(\ell _{2})^{\star }$ and have descriptions in some categories of topological vector spaces. But I am more or less sure that this was not stated up to now. Eozhik (talk) 15:05, 23 February 2020 (UTC)