{
  "id": "1502.00414",
  "version": "v1",
  "published": "2015-02-02T09:20:43.000Z",
  "updated": "2015-02-02T09:20:43.000Z",
  "title": "Concentration analysis in Banach spaces",
  "authors": [
    "Sergio Solimini",
    "Cyril Tintarev"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "The concept of a profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach-Alaoglu weak-star compactness theorem. We prove existence of profile decompositions for general bounded sequences in uniformly convex Banach spaces equipped with a group of bijective isometries, thus generalizing analogous results previously obtained for Sobolev spaces and for Hilbert spaces. Profile decompositions in uniformly convex Banach spaces are based on the notion of $\\Delta$-convergence by T. C. Lim instead of weak convergence, and the two modes coincide if and only if the norm satisfies the well-known Opial condition, in particular, in Hilbert spaces and $\\ell^{p}$-spaces, but not in $L^{p}(\\mathbb R^{N})$, $p\\neq2$. $\\Delta$-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies connection of $\\Delta$-convergence with Brezis-Lieb Lemma and gives a version of the latter without an assumption of convergence a.e.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-02T09:20:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "46B10",
      "46B50",
      "46B99"
    ],
    "keywords": [
      "concentration analysis",
      "uniformly convex banach spaces",
      "convergence",
      "profile decomposition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150200414S"
    }
  }
}