@article {
author = {Godefroy, Gilles and Lerner, Nicolas},
title = {Some natural subspaces and quotient spaces of $L^1$},
journal = {Advances in Operator Theory},
volume = {3},
number = {1},
pages = {61-74},
year = {2018},
publisher = {Tusi Mathematical Research Group (TMRG)},
issn = {2538-225X},
eissn = {2538-225X},
doi = {10.22034/aot.1702-1124},
abstract = {We show that the space $\text{Lip}_0(\mathbb R^n)$ is the dual space of $L^{1}({\mathbb R}^{n}; {\mathbb R}^{n})/N$ where $N$ is the subspace of $L^{1}({\mathbb R}^{n}; {\mathbb R}^{n})$ consisting of vector fields whose divergence vanishes identically. We prove that although the quotient space $L^{1}({\mathbb R}^{n}; {\mathbb R}^{n})/N$ is weakly sequentially complete, the subspace $N$ is not nicely placed - in other words, its unit ball is not closed for the topology $\tau_m$ of local convergence in measure. We prove that if $\Omega$ is a bounded open star-shaped subset of $\mathbb {R}^n$ and $X$ is a dilation-stable closed subspace of $L^1(\Omega)$ consisting of continuous functions, then the unit ball of $X$ is compact for the compact-open topology on $\Omega$. It follows in particular that such spaces $X$, when they have Grothendieck's approximation property, have unconditional finite-dimensional decompositions and are isomorphic to weak*-closed subspaces of $l^1$. Numerous examples are provided where such results apply.},
keywords = {nicely placed subspaces of $L^1$,Lipschitz-free spaces over $mathbb{R}^n$,subspaces of $l^1$},
url = {http://www.aot-math.org/article_44924.html},
eprint = {http://www.aot-math.org/article_44924_ba9c3c5c2f3766d6635df15b74db8914.pdf}
}