Timeline for Dual of Banach-valued $L^p$
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 29, 2015 at 13:09 | comment | added | Gerald Edgar | That Diestel--Uhl is a fine book. Have a look. | |
| Jan 29, 2015 at 13:08 | comment | added | András Bátkai | And for me a Banach space $E$ has the Radon-Nikodym property if and only if every Lipschitz function $f:[0,1]\to E$ is almost everywhere differentiable. | |
| Jan 29, 2015 at 13:06 | comment | added | Gerald Edgar | Here, we need only RNP for dual space $X^*$. Stegall's characterization: $X^*$ enjoys the RNP if and only if every separable subspace of $X$ has separable dual. | |
| Jan 29, 2015 at 12:18 | comment | added | Yemon Choi | @AndréHenriques just to add to Tomek's comment: X has RNP if and only if every bounded subset of X has the following geometric property ("dentability"): for each $r>0$ there is a point $x\in X$ that does not lie in the convex hull of $X\setminus$ (r-neighbourhood of $x$) | |
| Jan 29, 2015 at 12:03 | comment | added | Tomasz Kania | The Radon-Nikodym property is a property saying that the vector-valued version of the Radon-Nikodym theorem holds true. Conjecturally it is equivalent to saying that all convex, closed, bounded sets are closed convex hulls of their extreme points. | |
| Jan 29, 2015 at 11:14 | comment | added | André Henriques | What is the Radon-Nikodym property? Is it easy to say? | |
| Jan 28, 2015 at 18:54 | history | edited | Joonas Ilmavirta | CC BY-SA 3.0 |
Added a dollar.
|
| Jan 28, 2015 at 18:25 | comment | added | Gerald Edgar | Another important special case (where $X^*$ has the RNP) is when $X^*$ is separable. For example, if $X$ is separable and $X^*$ is not separable, we would expect the dual of $L^{p}(\mu, X)$ to include some non-Bochner-measurable functions. | |
| Jan 28, 2015 at 18:07 | history | edited | András Bátkai | CC BY-SA 3.0 |
added 89 characters in body
|
| Jan 28, 2015 at 18:04 | comment | added | András Bátkai | Yes, I add the details to the answer. Thank you. | |
| Jan 28, 2015 at 18:03 | comment | added | Yemon Choi | Just to clarify: presumably there is a missing quantifier, and the full result is: the dual has the desired form for all $\sigma$-finite $(\Omega,\mu)$ if and only if $X^*$ has RNP? | |
| Jan 28, 2015 at 18:01 | history | answered | András Bátkai | CC BY-SA 3.0 |