This is an announcement for the paper "Linear $q$--positive sets and their polar subspaces" by Stephen Simons. Abstract: In this paper, we define a Banach SNL space to be a Banach space with a certain linear map from it into its dual, and we develop the theory of $q$--positive linear subsets of Banach SNL spaces with Banach SNL dual spaces. We use this theory to give simplified proofs of some recent results of Bauschke, Borwein, Wang and Yao, and also of the classical Brezis–Browder theorem. Archive classification: math.FA Remarks: 11 pages Submitted from: simons@math.ucsb.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.0280 or http://arXiv.org/abs/1112.0280