Timeline for Unambiguous "weak" vector valued $L^{+\infty}$ spaces?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| May 26, 2013 at 10:27 | comment | added | TaQ | The basic problem is what would be the "right" way of defining the space $L^p(\mu,E)$ for $E$ a general locally convex space. If $E$ is a separable Banach(able locally convex) space with separable strong dual $E^{\prime}_\beta$, then obviously one has $L^{p'}(\mu,E^{\prime}_\sigma)=L^{p'}(\mu,E^{\prime}_\beta)$, i.e. the "weak" dual valued Lebesgue space is the classical Bochner space. | |
| May 26, 2013 at 10:26 | comment | added | TaQ | The purpose of the question is to try clarify whether it is unavoidably necessary to define vector valued Lebesgue spaces $L^p(\mu,F)$ for much more general locally convex spaces $F$ than just normable ones in order to be able to get suitable generalizations of the duality theory of scalar valued spaces. This problem has been adressed in the above mentioned Edwards' book but there are many restrictions in order for the representation of the dual of $L^p(\mu,E)$ as $L^{p'}(\mu,E^{\prime}_\sigma)$ to hold. | |
| May 26, 2013 at 10:25 | comment | added | TaQ | "Even if the dual of the separable Banach space is not separable, there is a perfectly respectable complete locally convex structure there for which the dual of $L^1(I,E)$ is canonically identifiable with the space of (equivalence classes of) of bounded, measurable functions--- the bounded, weak-star topology." Assuming that I understand the preceding as intended, so the dual of $L^1(I,E)$ would be represented by a (suitably defined) space $L^{+\infty}(I,E^{\prime}_\sigma)$. This is what has been in my mind already at the moment of posing the original question. | |
| May 26, 2013 at 8:19 | comment | added | jbc | By the way, with suitable tools, one can prove the required representation for the dual in a few lines. One can express $E$ as the closure of the union of an increasing sequence $(E_n)$ of finite dimensional subspaces. It follows easily that the required $L^1$ space is the inductive limit (in the category of Banach spaces with linear contractions as morphisms) of the $L^(I,E_n)$. The natural extension of $L^1$ duality theory to functions with values in finite dimensional spaces plus abstract nonsense on duality for the above category produces the stated identification of the dual. | |
| May 26, 2013 at 8:09 | comment | added | jbc | Even if the dual of the separable Banach space is not separable, there is a perfectly respectable complete locally convex structure there for which the dual of $L^1(I,E)$ is canonically identifiable with the space of (equivalence classes of) of bounded, measurable functions--- the bounded, weak-star topology. This not unimportant since probably the most significant example is that where $E$ is the space of bounded, linear operators on Hilbert space which is a non-separable dual of the nuclear operators. There this concept of measurability is ubiquitous in spectral theory. | |
| May 25, 2013 at 13:31 | comment | added | TaQ | ^ Thanks for the hint. I do not have the book of Diestel and Uhl at hand but can see the result by using Proposition 8.15.3 on page 575 in Edwards' book. | |
| May 25, 2013 at 5:45 | comment | added | Bill Johnson |
More generally, if $E'$ is separable, then weak$^*$ measurability into $E'$ gives strong measurability. This is in books; Diestel-Uhl comes to mind. It follows from the fact that the unit ball is weak$^*$ measurable when $E'$ is separable.
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| May 25, 2013 at 4:27 | history | edited | TaQ | CC BY-SA 3.0 |
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| Feb 12, 2012 at 16:41 | history | asked | TaQ | CC BY-SA 3.0 |