Timeline for finite codimension implies closed?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 15, 2011 at 17:35 | answer | added | Ramiro de la Vega | timeline score: 3 | |
| Sep 1, 2011 at 6:57 | vote | accept | Guy Relande | ||
| Sep 1, 2011 at 6:56 | vote | accept | Guy Relande | ||
| Sep 1, 2011 at 6:57 | |||||
| Aug 31, 2011 at 20:29 | answer | added | Bill Johnson | timeline score: 5 | |
| Aug 30, 2011 at 18:59 | comment | added | B R | @Bill Johnson: Silly of me, especially the second point (the first point was ignorance, which is rarely silly). I should have known to stay away from complementary subspaces . . . | |
| Aug 30, 2011 at 18:10 | comment | added | jjcale | The open mapping theorem holds for F-spaces (see Rudin, functional analysis). | |
| Aug 30, 2011 at 17:55 | comment | added | Bill Johnson |
@BR: You seem to be asserting that if a (finite dimensional) subspace has a closed topological complement, then every complement is closed, which is certainly not true even for Banach spaces. Also, the argument Guy mentions shows that there is no operator from $L^p$ to itself whose range is a proper, finite codimensional subspace.
|
|
| Aug 30, 2011 at 2:24 | comment | added | B R | Corollary 3 to Theorem 1 in Section 8.3 of Chapter 2 of Bourbaki's TVS states: "Let M be a finite dimensional subspace of a Hausdorff locally convex tvs E. Then there exists a closed vector subspace N of E that is a topological complement of N." So local convexity (in particular, the Hahn-Banach theorem) is sufficient. Exercise 11 in Chapter 1 of Rudin's Functional Analysis is to prove that all subspaces of L^p (0<p<1) of finite codimension are dense. So some hypothesis is necessary. | |
| Aug 29, 2011 at 19:46 | comment | added | Guy Relande | It just means that you have a complementary subspace of finite dimension or that the quotient space is finite dimensional (not necessarily separated). | |
| Aug 29, 2011 at 19:09 | comment | added | Thierry Zell | I'm a little bit naive: can you clarify what the correct definition of finite codimension is in this specific context? Thanks! | |
| Aug 29, 2011 at 18:55 | comment | added | kostja | For banach spaces, please see this question: mathoverflow.net/questions/54981 | |
| Aug 29, 2011 at 18:37 | history | edited | Yemon Choi |
added FA tag
|
|
| Aug 29, 2011 at 18:32 | history | edited | David White |
edited tags
|
|
| Aug 29, 2011 at 18:13 | history | asked | Guy Relande | CC BY-SA 3.0 |