1,482 research outputs found

    Geometry of Banach spaces and biorthogonal systems

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    A separable Banach space X contains â„“1\ell_1 isomorphically if and only if X has a bounded wc_0^*-stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded wc_0^*-biorthogonal system

    On the size of approximately convex sets in normed spaces

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    Let X be a normed space. A subset A of X is approximately convex if d(ta+(1−t)b,A)≤1d(ta+(1-t)b,A) \le 1 for all a,b∈Aa,b \in A and t∈[0,1]t \in [0,1] where d(x,A)d(x,A) is the distance of xx to AA. Let \Co(A) be the convex hull and \diam(A) the diameter of AA. We prove that every nn-dimensional normed space contains approximately convex sets AA with \mathcal{H}(A,\Co(A))\ge \log_2n-1 and \diam(A) \le C\sqrt n(\ln n)^2, where H\mathcal{H} denotes the Hausdorff distance. These estimates are reasonably sharp. For every D>0D>0, we construct worst possible approximately convex sets in C[0,1]C[0,1] such that \mathcal{H}(A,\Co(A))=\diam(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.Comment: 32 pages. See also http://www.math.sc.edu/~howard
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