Timeline for Can $L^1_{loc}$ be represented as colimit?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 13, 2020 at 13:39 | history | edited | ABIM | CC BY-SA 4.0 |
added 103 characters in body
|
| S Dec 9, 2019 at 14:26 | history | bounty ended | ABIM | ||
| S Dec 9, 2019 at 14:26 | history | notice removed | ABIM | ||
| S Dec 3, 2019 at 7:31 | history | bounty started | ABIM | ||
| S Dec 3, 2019 at 7:31 | history | notice added | ABIM | Authoritative reference needed | |
| Dec 1, 2019 at 17:20 | vote | accept | ABIM | ||
| Dec 1, 2019 at 15:39 | comment | added | Dmitri Pavlov | @JochenWengenroth: The original question (before it was edited) clearly talked about a set L^1_loc, so the colimit was assumed to be in the category of sets. Now that the question was retroactively edited, my answer no longer makes sense. | |
| Dec 1, 2019 at 14:13 | answer | added | Jochen Wengenroth | timeline score: 6 | |
| Dec 1, 2019 at 13:04 | comment | added | ABIM | @NeilStrickland I added a note. | |
| Dec 1, 2019 at 13:04 | history | edited | ABIM | CC BY-SA 4.0 |
added 310 characters in body
|
| Dec 1, 2019 at 12:30 | comment | added | Neil Strickland | "colimit" only makes sense in the context of a category, and there are various different categories that you could be using. What are your objects and morphisms? | |
| Dec 1, 2019 at 11:57 | comment | added | ABIM | @JochenWengenroth I would expect $\operatorname{colim} L^1_{m_K}$ (essentially definition almost) to be at-least as fine as $L^1_{loc}$, but why ("how much") finer? From your argument I see that it should be finer, but is there a concrete example of [functions] converging in $L^1_{loc}$ but not in $\operatorname{colim}L^1_{m_K}$, for example? | |
| Dec 1, 2019 at 11:38 | history | edited | ABIM | CC BY-SA 4.0 |
deleted 86 characters in body
|
| Dec 1, 2019 at 10:30 | comment | added | Jochen Wengenroth | @DmitriPavlov If $L^1_{m_K}$ is the space of $L^1$-functions with support in $K$ then we have an inductive spectrum of spaces and hence an inductive limit (which is the same as a colimit). The corresponding topology in TOP (and also in the category LCS of locally convex spaces) is much finer than the Frechet topology of $L^1_{loc}$ (which is the reverse or projective limit in LCS with respect to the restriction mappings). | |
| Dec 1, 2019 at 0:13 | comment | added | Dmitri Pavlov | If you change the definition of L^1_{m_K} to say that it is a subspace of L^1 consisting of functions supported on K, the resulting colimit is isomorphic to L^1_loc, pretty much by definition of L^1_loc. | |
| Nov 30, 2019 at 23:48 | comment | added | Dmitri Pavlov | Using the standard definition of L^1, a function in L^1_{m_K} can take arbitrary values outside of K. Thus L^1_{m_K1} is not a subset of L^1_{m_K2} for K1⊂K2. | |
| Nov 30, 2019 at 18:39 | history | asked | ABIM | CC BY-SA 4.0 |