Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Some inequalities in functional analysis, combinatorics, and probability theory. | Some inequalities in functional analysis combinatorics and probability theory Chunrong Feng Liangpan Li Jian Shen Submitted Aug 21 2009 Accepted Mar 30 2010 Published Apr 5 2010 Mathematics Subject Classification 46C05 05A20 60C05 11T99 Abstract The main purpose of this paper is to show that many inequalities in functional analysis probability theory and combinatorics are immediate corollaries of the best approximation theorem in inner product spaces. Besides as applications of the de Caen-Selberg inequality the finite field Kakeya and Nikodym problems are also studied. Keywords inner product space orthogonal projection Kakeya set Nikodym set 1 Brief Introduction Let H be an inner product space over R throughout. Given x G H and a finite dimensional subspace M denote by xM the orthogonal projection of x onto M. It is geometrically evident that we always assume 0 0 in this paper . 2 2 l xh2 xmII2 max------------------ max . 1 yeM y 2 yeM y 2 Particularly if M spanỊyq for some given set of elements yi . yn then HxH2 max ai . an eRn x Ebi aiyi 2 II Eli ayiW2 2 Department of Mathematics Shanghai Jiao Tong University Shanghai 200240 China Department of Mathematical Sciences Loughborough University Leics LE11 3TU UK. E-mail fcr@. Research was supported by the Mathematical Tianyuan Foundation of China No. 10826090 . Department of Mathematics Shanghai Jiao Tong University Shanghai 200240 China. E-mail lil-iangpan@. Research was supported by the Mathematical Tianyuan Foundation of China No. 10826088 . Department of Mathematics Texas State University San Marcos TX 78666 USA. E-mail js48@. Research was supported by NSF CNS 0835834 and Texas Higher Education Coordinating Board ARP 003615-0039-2007 . THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R58 1 The main purpose of this paper is to show that many inequalities in functional analysis probability theory and combinatorics are immediate corollaries of 2 . For the sake of completeness we .
