Some Natural Subspaces and Quotient Spaces of L
نویسنده
چکیده
We show that the space Lip0(R) is the dual space of L(R;R)/N where N is the subspace of L(R;R) consisting of vector fields whose divergence vanishes identically. We prove that although the quotient space L(R;R)/N is weakly sequentially complete, the subspace N is not nicely placed in other words, its unit ball is not closed for the topology τm of local convergence in measure. We prove that if Ω is a bounded open star-shaped subset of R and X is a dilation-stable closed subspace of L(Ω) consisting of continuous functions, then the unit ball of X is compact for the compact-open topology on Ω. 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. Numerous examples are provided where such results apply.
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