Functional Analysis and Infinite-Dimensional Geometry

By Marian J. Fabian, Petr Habala, P. Hájek, Vicente Montesinos Santalucía, J. Pelant, V. Zizler. 2001

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The spatial distribution in infinite dimensional spaces and related quantiles and depths (2014, methodology): “We refer to [14] for an exposition on the theory of Banach spaces, and [3] for the relevant details on convex analysis in Banach spaces.”
Algebras in sets of queer functions (2007, positive): “Using a result of Sierpinski (see [11] for example), let {Nα} be a family of 2א0 almost disjoint infinite subsets of N (i.”
Operators That Achieve the Norm (2010, methodology): “Notation and background At this point we fix the functional notation used in this paper, and recall some well known results from function analysis, see [3], [6].”
Reproducing kernel Banach spaces for machine learning (2009, methodology): “Thirdly, RKHS methods require symmetric kernels, which might be violated (1) (2)f(y) == (f(·), K(y, ·))rt, is hermitian (especially symmetric, when K is real-valued) and positive semi-definite.”
Coefficient quantization for frames in Banach spaces (2007, neutral): “[11]) to the property that for all (ai) ∈ c00 Ku = sup {∥∥∥∑±ai xi∥∥∥: ∥∥∥∑ai xi∥∥∥= 1}< ∞.”
COMONOTONIC MEASURES OF MULTIVARIATE RISKS (2012, neutral): “By a celebrated theorem of Bishop and Phelps (see again Fabian et al. 2001), there is a dense subset H of L2 (in fact, a dense Gδ) such that, for every X ∈ H, the maximum of 〈 X∗, X 〉 for X∗ ∈ C is attained at a strongly exposed point.”