Timeline for Is there a version of Fischer-Riesz theorem for Banach space?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jan 5, 2021 at 14:36 | comment | added | Pietro Majer | One usually distinguishes between $\mathcal{L}^p(\Omega,B)$ (a space of measurable functions, what you denote $L^p(\Omega,B)$, and $L^p(\Omega,B)$, its quotient modulo a.e. equality of functions, thus its element are classes of functions, not functions $\Omega\to B$. Strictly speaking $\mathcal{L}^p(\Omega,B)$ fails to be a Banach space just because $\||\cdot\||$ is not a norm (just a seminorm), whenever $(\Omega, \mathcal{F}, P)$ has non-empty set of null measure. | |
| Oct 26, 2018 at 13:50 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
corrected typo in the title + added a top-level tag
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| Oct 25, 2018 at 23:11 | answer | added | Gerald Edgar | timeline score: 14 | |
| Oct 25, 2018 at 21:48 | comment | added | Jochen Glueck | As the issue pointed out by @GeraldEdgar shows, the "right" definition of measurable vector-valued functions is not via the Boral $\sigma$-algebra on $B$ but via approximation by simple functions. See for instance Section 1.1 of the book by Hytönen et. al. quoted in my comment below. | |
| Oct 25, 2018 at 18:49 | comment | added | Gerald Edgar | NO! Unless $B$ is separable, or $(\Omega, F, P)$ is a special sort of measurable space, this can fail. In general, if you restrict to functions with almost all values in a separable subspace of $B$, then you get the Bochner spaces, which are, indeed, complete. | |
| Oct 25, 2018 at 18:00 | answer | added | Piotr Hajlasz | timeline score: 8 | |
| Oct 25, 2018 at 17:02 | comment | added | Tomasz Kania | You may be interested also in this book springer.com/gp/book/9783540637455 | |
| Oct 25, 2018 at 16:50 | answer | added | Nate Eldredge | timeline score: 7 | |
| Oct 25, 2018 at 16:39 | history | asked | Taro Tokyo | CC BY-SA 4.0 |