Timeline for Duality relations for Lebesgue spaces of sections of vector bundles
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| Aug 7, 2013 at 9:39 | comment | added | user14166 | by my setting, i just mean the setting of having a non-trivial bundle rather than a trivial bundle. even with your additional assumptions, i can't get Dudley's argument to work - although we can write the space as an increasing union of open sets, the bundle won't necessarily be trivial over these sets. we can get the required bound for $y$ on trivialising open sets, but we don't automatically have it on unions. it's possible that i've misunderstood something in your argument - my apologies if this is the case! i would certainly appreciate more detail, if you have the time. | |
| Aug 7, 2013 at 8:59 | comment | added | TaQ | @ Alex Amenta: Is your setting such that the additional assumptions given at the beginning of my answer are not satisfied? Have you looked at the reference I gave? If you accept my additional assumptions, then it is only an exercise level task to reformulate Dudley's argument in your situation. If you still do not understand it, I can add the details to my answer when I have better time. | |
| Aug 7, 2013 at 1:24 | comment | added | user14166 | in my setting, we don't have the above $L^{p^\prime}$ bound on all sets $U$ - we only know it on trivialising open sets. even though the space is $\sigma$-finite, we generally can't write the space as an increasing union of trivialising open sets. | |
| Aug 7, 2013 at 1:22 | comment | added | user14166 | here is where the issue lies with this approach: in the final approximation step, one must take an increasing sequence of sets $E(n)$ of finite measure such that $\cup_n E(n) = X$. in the usual setting, we know that the function $y$ induced by $u \in (L^p)^*$ satisfies $||y||_{L^p(U)} \leq ||u||$ for all sets $U$ of finite measure - in particular, we can take $U = E(n)$. (continued below) | |
| Aug 7, 2013 at 0:47 | comment | added | TaQ | This argument readily generalizes to the case of sections of a finite-rank (or even Hilbert space valued) vector bundle under the additional assumptions on the base topological space given at the beginning of my answer. | |
| Aug 7, 2013 at 0:46 | comment | added | TaQ | @Alex Amenta: The inequality $\|y\|_{p'}^{\,p'}\le\|u\|\,\|y\|_{p'}^{\,\frac{p'}p}$ only gives the basic idea behind the actual argument for which I refer you to see e.g. page 163 in the proof of the Riesz Representation Theorem 6.4.1 in Richard M. Dudley's book "Real Analysis and Probability" (Wadsworth, 1989; there is also a later edition with possibly different numberings) where the exact measure theoretic details are given in the case of scalar valued functions. | |
| Aug 6, 2013 at 23:45 | comment | added | user14166 | i don't understand the last part of your proof - how do you choose $x$ such that $|u(x)| \leq ||u|| ||x||_p$ turns into the inequality for $||y||_{p^\prime}$? | |
| Aug 6, 2013 at 21:31 | history | edited | TaQ | CC BY-SA 3.0 |
added 70 characters in body
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| Aug 6, 2013 at 21:13 | history | answered | TaQ | CC BY-SA 3.0 |