{
  "id": "1407.1100",
  "version": "v2",
  "published": "2014-07-04T00:48:45.000Z",
  "updated": "2014-10-27T03:55:06.000Z",
  "title": "$r_L$-density and maximal monotonicity",
  "authors": [
    "Stephen Simons"
  ],
  "comment": "41 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We discuss \"Banach SN spaces\", which include Hilbert spaces, negative Hilbert spaces, and the product of any real Banach space with its dual. We introduce \"L-positive\" sets, which generalize monotone multifunctions from a Banach space into its dual. We introduce the concept of $r_L$-density: the closed $r_L$-dense monotone multifunctions from a Banach space into its dual form a strict subset of the maximally monotone ones. We give two short proofs that the subdifferential of a proper convex lower semicontinuous function on a Banach space is $r_L$-dense. We give sum theorems for closed monotone $r_L$-dense multifunctions under very general constraint conditions, and we prove that any closed monotone $r_L$-dense multifunction is of type (ANA), strongly maximally monotone, of type (FPV), of type (FP), of type (NI), and has a number of other very desirable properties. In the (FP) and (NI) case the converse results are true.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-04T00:48:45.000Z",
      "title": "A \"density\" and maximal monotonicity",
      "abstract": "We discuss \"Banach SN spaces\", which include Hilbert spaces, negative Hilbert spaces, and the product of any real Banach space with its dual. We introduce \"L-positive\" sets, which generalize monotone multifunctions from a Banach space into its dual. We introduce a new \"density\" concept: the closed \"dense\" monotone multifunctions from a Banach space into its dual form a strict subset of the maximally monotone ones. We give a very short proof that the subdifferential of a proper convex lower semicontinuous function on a Banach space has this \"density\" property. We give sum theorems for closed monotone \"dense\" multifunctions under very general constraint conditions, and we prove that any closed monotone \"dense\" multifunction is of type (ANA), strongly maximally monotone, of type (FPV), of type (FP), of type (NI), and has a number of other very desirable properties.",
      "comment": "30 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-27T03:55:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47H05",
      "47N10",
      "52A41",
      "46A20"
    ],
    "keywords": [
      "maximal monotonicity",
      "proper convex lower semicontinuous function",
      "monotone multifunctions",
      "hilbert spaces",
      "maximally monotone"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.1100S"
    }
  }
}