Timeline for Extending Continuous Sublinear maps on dense subsets of a Banach space
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 6, 2019 at 19:10 | answer | added | LinearOperator32 | timeline score: 0 | |
| May 12, 2014 at 21:14 | history | edited | Ricardo Andrade |
edited tags; replaced deprecated tag 'topology'
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| May 2, 2011 at 1:08 | answer | added | Ady | timeline score: 1 | |
| Apr 20, 2011 at 3:02 | comment | added | Yemon Choi | When you say: $T:X\to Y$ is continuous - what topology is one putting on $X$? The subspace topology inherited from $X'$ ? | |
| Apr 19, 2011 at 4:09 | history | edited | Gerald Edgar | CC BY-SA 3.0 |
added 9 characters in body
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| Apr 19, 2011 at 4:08 | comment | added | Gerald Edgar | edited to show this | |
| Apr 19, 2011 at 1:22 | comment | added | Jeffrey | My last comment was a response to Yemon, but this one is a response to Bill: X is a subspace. That is, it is a normed linear space. Hence, it is closed under addition. Sorry if I was not clear and you thought I meant topological space. | |
| Apr 19, 2011 at 1:20 | comment | added | Jeffrey | Yes. The purpose of proving the theorem above is to be able to use the Marcinkiewicz Interpolation theorem. The interpolation theorem states that if T is subadditive, and operates on functions with support of finite measure, then inequalities concerning infinity norms and weak L1 norms imply inequalities concerning p norms for 1<p<infinity. The problem is getting something useful out of this theorem for functions that are not bounded, supported on sets of finite measure. To still have the bounds would be nice, since it would imply a maximal inequality for the Hardy Littlewood function. | |
| Apr 19, 2011 at 1:03 | comment | added | Bill Johnson | As stated, there are trivial counterexamples even when both spaces are the real line: Take $X$ to be a dense subset such that $x+y$ is not in $X$ for all $x$, $y$ in $X4 to make (1) irrelevant. | |
| Apr 18, 2011 at 23:52 | comment | added | Yemon Choi | Do you really want assume continuity of $T$ in your second sentence? | |
| Apr 18, 2011 at 22:47 | history | asked | Jeffrey | CC BY-SA 3.0 |