%0 Journal Article
%T Some natural subspaces and quotient spaces of $L^1$
%J Advances in Operator Theory
%I Tusi Mathematical Research Group (TMRG)
%Z 2538-225X
%A Godefroy, Gilles
%A Lerner, Nicolas
%D 2018
%\ 01/01/2018
%V 3
%N 1
%P 61-74
%! Some natural subspaces and quotient spaces of $L^1$
%K nicely placed subspaces of $L^1$
%K Lipschitz-free spaces over $mathbb{R}^n$
%K subspaces of $l^1$
%R 10.22034/aot.1702-1124
%X 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.
%U http://www.aot-math.org/article_44924_ba9c3c5c2f3766d6635df15b74db8914.pdf