TY - JOUR
ID - 44924
T1 - Some natural subspaces and quotient spaces of $L^1$
JO - Advances in Operator Theory
JA - AOT
LA - en
SN -
A1 - Godefroy, Gilles
A1 - Lerner, Nicolas
Y1 - 2018
PY - 2018/01/01
VL - 3
IS - 1
SP - 61
EP - 74
KW - nicely placed subspaces of $L^1$
KW - Lipschitz-free spaces over $mathbb{R}^n$
KW - subspaces of $l^1$
DO - 10.22034/aot.1702-1124
N2 - 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.
UR - http://www.aot-math.org/article_44924.html
L1 - http://www.aot-math.org/pdf_44924_ba9c3c5c2f3766d6635df15b74db8914.html
ER -