Journal ArticleDOI
Invertibility of ‘large’ submatrices with applications to the geometry of Banach spaces and harmonic analysis
Jean Bourgain,L. Tzafriri +1 more
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In this paper, the main problem of restricted invertibility of linear operators acting on finite dimensionallp-spaces is investigated, and the results obtained below enable us to complete earlier work on the structure of complemented subspaces of lp-space which have extremal euclidean distance.Abstract:
The main problem investigated in this paper is that of restricted invertibility of linear operators acting on finite dimensionallp-spaces. Our initial motivation to study such questions lies in their applications. The results obtained below enable us to complete earlier work on the structure of complemented subspaces ofLp-spaces which have extremal euclidean distance.read more
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References
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Book ChapterDOI
On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.
Journal ArticleDOI
On the density of families of sets
TL;DR: This paper will answer the question in the affirmative by determining the exact upper bound of T if T is a family of subsets of some infinite set S then either there exists to each number n a set A ⊂ S with |A| = n such that |T ∩ A| = 2n or there exists some number N such that •A| c for each A⩾ N and some constant c.
Journal ArticleDOI
Probability Inequalities for the Sum of Independent Random Variables
TL;DR: In this article, a number of inequalities which improve on existing upper limits to the probability distribution of the sum of independent random variables are presented, which are applicable when the number of component random variables is small and/or have different distributions.