{
  "id": "1305.4186",
  "version": "v1",
  "published": "2013-05-17T20:13:29.000Z",
  "updated": "2013-05-17T20:13:29.000Z",
  "title": "Some combinatorial principles for trees and applications to tree-families in Banach spaces",
  "authors": [
    "Costas Poulios",
    "Athanasios Tsarpalias"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Suppose that $(x_s)_{s\\in S}$ is a normalized family in a Banach space indexed by the dyadic tree $S$. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree-families. More precisely, assuming that for any infinite chain $\\beta$ of $S$ the sequence $(x_s)_{s\\in\\beta}$ is weakly null, we prove that there exists a subtree $T$ of $S$ such that for any infinite chain $\\beta$ of $T$ the sequence $(x_s)_{s\\in\\beta}$ is nearly (resp., convexly) unconditional. In the case where $(f_s)_{s\\in S}$ is a family of continuous functions, under some additional assumptions, we prove the existence of a subtree $T$ of $S$ such that for any infinite chain $\\beta$ of $T$, the sequence $(f_s)_{s\\in\\beta}$ is unconditional. Finally, in the more general setting where for any chain $\\beta$, $(x_s)_{s\\in\\beta}$ is a Schauder basic sequence, we obtain a dichotomy result concerning the semi-boundedly completeness of the sequences $(x_s)_{s\\in\\beta}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-17T20:13:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10",
      "46B15"
    ],
    "keywords": [
      "banach space",
      "combinatorial principles",
      "infinite chain",
      "tree-families",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.4186P"
    }
  }
}